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  • Articles  (112)
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  • Global optimization  (65)
  • Optimal control  (47)
  • Springer  (112)
  • 2020-2022
  • 1995-1999  (112)
  • Mathematics  (112)
  • 1
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    A Technique for Stochastic Control Problems with Unbounded Control Set (1999)
    Springer
    Journal of theoretical probability 12 (1999), S. 255-270 
    Details
    ISSN: 1572-9230
    Keywords: Optimal control ; stochastic differential equations ; convergence in law ; unbounded control set ; suboptimal control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We describe a change of time technique for stochastic control problems with unbounded control set. We demonstrate the technique on a class of maximization problems that do not have optimal controls. Given such a problem, we introduce an extended problem which has the same value function as the original problem and for which there exist optimal controls that are expressible in simple terms. This device yields a natural sequence of suboptimal controls for the original problem. By this we mean a sequence of controls for which the payoff functions approach the value function.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1021761030407
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  • 2
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    An adaptive stochastic global optimization algorithm for one-dimensional functions (1995)
    Springer
    Annals of operations research 58 (1995), S. 261-278 
    Details
    ISSN: 1572-9338
    Keywords: Global optimization ; Wiener process ; sequential stopping rules ; one-step look-ahead
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract In this paper a new algorithm is proposed, based upon the idea of modeling the objective function of a global optimization problem as a sample path from a Wiener process. Unlike previous work in this field, in the proposed model the parameter of the Wiener process is considered as a random variable whose conditional (posterior) distribution function is updated on-line. Stopping criteria for Bayesian algorithms are discussed and detailed proofs on finite-time stopping are provided.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02096402
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  • 3
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    Optimal budgetary and monetary policies under uncertainty: A stochastic control approach (1995)
    Springer
    Annals of operations research 58 (1995), S. 379-402 
    Details
    ISSN: 1572-9338
    Keywords: Optimal control ; stochastic control ; dynamic systems ; economics ; public-sector applications ; optimization ; budgetary policies ; monetary policy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract In this paper, we determine optimal budgetary and monetary policies for Austria using a small macroeconometric model. We use a Keynesian model of the Austrian economy, called FINPOL1, estimated by ordinary least squares, which relates the main objective variables of Austrian economic policies, such as the growth rate of real gross domestic product, the rate of unemployment, the rate of inflation, the balance of payments, and the ratio of the federal budget deficit to GDP, to fiscal and monetary policy instruments, namely expenditures and revenues of the federal budget and money supply. Optimal fiscal and monetary policies are calculated for the model under a quadratic objective function using the algorithm OPTCON for the optimum control of nonlinear stochastic dynamic systems. Several control experiments are performed in order to assess the influence of different kinds of uncertainty on optimal budgetary and monetary policies. Apart from deterministic optimization runs, different assumptions about parameter uncertainties are introduced; the results of these different stochastic optimum control experiments are compared and interpreted.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02038862
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  • 4
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    Sensitivity, controllability, and necessary conditions of optimal control problems governed by integral equations (1995)
    Springer
    Applied mathematics & optimization 32 (1995), S. 73-97 
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    ISSN: 1432-0606
    Keywords: Optimal control ; State constraint ; Volterra integral equation ; Necessary conditions ; Controllability ; Sensitivity ; Value function ; Proximal normal analysis ; Proximal analysis ; 49J22 ; 49K22 ; 93B05 ; 93B06 ; 93B35
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Infinite-dimensional perturbations in all constraints of an optimal control problem governed by a Volterra integral equation with the presence of a state constraint are considered. These perturbations give rise to a value function, whose analysis through the proximal normal technique provides sensitivity, controllability, and even necessary conditions for the basic problem. Actually all information about the value function is contained in Clarke's normal cone of its epigraph, which can be characterized by the proximal normal formula.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01189904
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  • 5
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    Value function and optimality condition for semilinear control problems. II: Parabolic case (1996)
    Springer
    Applied mathematics & optimization 33 (1996), S. 1-33 
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    ISSN: 1432-0606
    Keywords: Optimal control ; Value function ; Analytic semigroups ; Semiconcave functions ; Distributed parameter systems ; 49K20 ; 49L20 ; 47B44 ; 49J52
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we continue to study properties of the value function and of optimal solutions of a semilinear Mayer problem in infinite dimensions. Applications concern systems governed by a state equation of parabolic type. In particular, the issues of the joint Lipschitz continuity and semiconcavity of the value function are treated in order to investigate the differentiability of the value function along optimal trajectories.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01187960
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  • 6
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    Pure adaptive search for finite global optimization (1995)
    Springer
    Mathematical programming 69 (1995), S. 443-448 
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    ISSN: 1436-4646
    Keywords: Global optimization ; Discrete optimization ; Algorithm complexity ; Random search ; Markov chains
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Pure Adaptive Search is a stochastic algorithm which has been analyzed for continuous global optimization. When a uniform distribution is used in PAS, it has been shown to have complexity which is linear in dimension. We define strong and weak variations of PAS in the setting of finite global optimization and prove analogous results. In particular, for then-dimensional lattice {1,⋯,k} n , the expected number of iterations to find the global optimum is linear inn. Many discrete combinatorial optimization problems, although having intractably large domains, have quite small ranges. The strong version of PAS for all problems, and the weak version of PAS for a limited class of problems, has complexity the order of the size of the range.
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    URL: http://dx.doi.org/10.1007/BF01585570
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  • 7
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    Global one-dimensional optimization using smooth auxiliary functions (1998)
    Springer
    Mathematical programming 81 (1998), S. 127-146 
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    ISSN: 1436-4646
    Keywords: Global optimization ; Lipschitzean first derivatives ; Numerical algorithms ; Convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper new global optimization algorithms are proposed for solving problems where the objective function is univariate and has Lipschitzean first derivatives. To solve this problem, smooth auxiliary functions, which are adaptively improved during the course of the search, are constructed. Three new algorithms are introduced: the first used the exact a priori known Lipschitz constant for derivatives; the second, when this constant is unknown, estimates it during the course of the search and finally, the last method uses neither the exact global Lipschitz constant nor its estimate but instead adaptively estimates the local Lipschitz constants in different sectors of the search region during the course of optimization. Convergence conditions of the methods are investigated from a general viewpoint and some numerical results are also given. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584848
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  • 8
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    The optimal control approach to generalized multiprocessor scheduling (1996)
    Springer
    Algorithmica 15 (1996), S. 17-49 
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    ISSN: 1432-0541
    Keywords: Scheduling ; Multiprocessor scheduling ; Parallel algorithms ; NP-completeness ; Theoretical computer science ; Operations research ; Optimal control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we present several new results in the theory of homogeneous multiprocessor scheduling. We start with some assumptions about the behavior of tasks, with associated precedence constraints, as processor power is applied. We assume that as more processors are applied to a task, the time taken to compute it decreases, yielding some speedup. Because of communication, synchronization, and task scheduling overhead, this speedup increases less than linearly with the number of processors applied. We also assume that the number of processors which can be assigned to a task is a continuous variable, with a view to exploiting continuous mathematics. The optimal scheduling problem is to determine the number of processors assigned to each task, and task sequencing, to minimize the finishing time. These assumptions allow us to recast the optimal scheduling problem in a form which can be addressed by optimal control theory. Various theorems can be proven which characterize the optimal scheduling solution. Most importantly, for the special case where the speedup function of each task isp α , wherep is the amount of processing power applied to the task, we can directly solve our equations for the optimal solution. In this case, for task graphs formed from parallel and series connections, the solution can be derived by inspection. The solution can also be shown to be shortest path from the initial to the final state, as measured by anl 1/α distance metric, subject to obstacle constraints imposed by the precedence constraints.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01942605
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  • 9
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    Provably good approximation algorithms for optimal kinodynamic planning: Robots with decoupled dynamics bounds (1995)
    Springer
    Algorithmica 14 (1995), S. 443-479 
    Details
    ISSN: 1432-0541
    Keywords: Robot motion planning ; Optimal control ; Polynomial-timeɛ-approximation algorithm ; Time-optimal trajectory ; Shortest path ; Kinodynamics ; Polyhedral obstacles
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider the following problem: given a robot system, find a minimal-time trajectory that goes from a start state to a goal state while avoiding obstacles by a speed-dependent safety margin and respecting dynamics bounds. In [1] we developed a provably good approximation algorithm for the minimum-time trajectory problem for a robot system with decoupled dynamics bounds (e.g., a point robot in ℝ3). This algorithm differs from previous work in three ways. It is possible (1) to bound the goodness of the approximation by an error termɛ; (2) to bound the computational complexity of our algorithm polynomially; and (3) to express the complexity as a polynomial function of the error term. Hence, given the geometric obstacles, dynamics bounds, and the error termɛ, the algorithm returns a solution that isɛ-close to optimal and requires only a polynomial (in (1/ɛ)) amount of time. We extend the results of [1] in two ways. First, we modify it to halve the exponent in the polynomial bounds from 6d to 3d, so that the new algorithm isO(c d N 1/ɛ)3d ), whereN is the geometric complexity of the obstacles andc is a robot-dependent constant. Second, the new algorithm finds a trajectory that matches the optimal in time with anɛ factor sacrificed in the obstacle-avoidance safety margin. Similar results hold for polyhedral Cartesian manipulators in polyhedral environments. The new results indicate that an implementation of the algorithm could be reasonable, and a preliminary implementation has been done for the planar case.
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    URL: http://dx.doi.org/10.1007/BF01586636
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  • 10
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    Provably good approximation algorithms for optimal kinodynamic planning for Cartesian robots and open-chain manipulators (1995)
    Springer
    Algorithmica 14 (1995), S. 480-530 
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    ISSN: 1432-0541
    Keywords: Robot motion planning ; Optimal control ; Polynomial-timeɛ-approximation algorithm ; Time-optimal trajectory ; Full dynamics ; Shortest path ; Kinodynamics ; Polyhedral obstacles
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Inoptimal kinodynamic planning, given a robot system, we must find a minimal-time trajectory that goes from a start state to a goal state while avoiding obstacles by a speed-dependent safety margin and respecting dynamics bounds. With Canny and Reif [1], we approached this problem from anɛ-approximation standpoint and introduced a provably good approximation algorithm for optimal kinodynamic planning for a robot obeying particle dynamics. If a solution exists, this algorithm returns a trajectoryɛ-close to optimal in time polynomial in both (1/ɛ) and the geometric complexity. We extend [1] and [2] tod-link three-dimensional robots with full rigid-body dynamics amidst obstacles. Specifically, we describe polynomial-time approximation algorithms for Cartesian robots obeyingL 2 dynamic bounds and for open-kinematic-chain manipulators with revolute and prismatic joints. The latter class includes many industrial manipulators. The correctness and complexity of these algorithms rely on new trajectory tracking lemmas for robots with coupled dynamics bounds.
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    URL: http://dx.doi.org/10.1007/BF01586637
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  • 11
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    On unbounded solutions of Bellman's equation associated with optimal switching control problems with state constraints (1995)
    Springer
    Applied mathematics & optimization 31 (1995), S. 1-17 
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    ISSN: 1432-0606
    Keywords: Bellman equation ; Hamilton-Jacobi equation ; Quasi-variational inequality ; Viscosity solution ; Unbounded solutions ; Optimal switching ; Optimal control ; State constraints ; Production engineering ; 49C05 ; 49C20 ; 49B10 ; 49A36
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study a quasi-variational inequality system with unbounded solutions. It represents the Bellman equation associated with an optimal switching control problem with state constraints arising from production engineering. We show that the optimal cost is the unique viscosity solution of the system.
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    URL: http://dx.doi.org/10.1007/BF01182554
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  • 12
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    Constrained retarded nonlinear optimal control problems in Banach state space (1996)
    Springer
    Applied mathematics & optimization 34 (1996), S. 191-230 
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    ISSN: 1432-0606
    Keywords: Optimal control ; Retarded systems ; Control in infinite-dimensional state space ; Primary 49K25 ; Secondary 34K35
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Necessary conditions are proved for the optimal control of solutions of ordinary and retarded differential equations in a Banach state space, with mixed and pure state restrictions. The treatment includes the possibility of point targets. A generalization of earlier results for finite-dimensional or Hilbert state spaces is obtained.
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    URL: http://dx.doi.org/10.1007/BF01182624
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  • 13
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    Analytical solutions to the optimization of a quadratic cost function subject to linear and quadratic equality constraints (1996)
    Springer
    Applied mathematics & optimization 34 (1996), S. 161-182 
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    ISSN: 1432-0606
    Keywords: Global optimization ; Quadratic cost ; Quadratic equality constraints ; Multivariate polynomials ; Resultants ; 49N99 ; 90C26 ; 90C30 ; 65K05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In the area of broad-band antenna array signal processing, the global minimum of a quadratic equality constrained quadratic cost minimization problem is often required. The problem posed is usually characterized by a large optimization space (around 50–90 tuples), a large number of linear equality constraints, and a few quadratic equality constraints each having very low rank quadratic constraint matrices. Two main difficulties arise in this class of problem. Firstly, the feasibility region is nonconvex and multiple local minima abound. This makes conventional numerical search techniques unattractive as they are unable to locate the global optimum consistently (unless a finite search area is specified). Secondly, the large optimization space makes the use of decision-method algorithms for the theory of the reals unattractive. This is because these algorithms involve the solution of the roots of univariate polynomials of order to the square of the optimization space. In this paper we present a new algorithm which exploits the structure of the constraints to reduce the optimization space to a more manageable size. The new algorithm relies on linear-algebra concepts, basic optimization theory, and a multivariate polynomial root-solving tool often used by decision-method algorithms.
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    URL: http://dx.doi.org/10.1007/BF01182622
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  • 14
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    Boundary optimal control of MHD flows (1995)
    Springer
    Applied mathematics & optimization 32 (1995), S. 143-162 
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    ISSN: 1432-0606
    Keywords: Optimal control ; Magnetohydrodynamics ; Electrically conducting fluid ; Primary 49J20 ; 49K20 ; Secondary 35Q35 ; 49M05 ; 76W05 ; 93C10 ; 93C20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper deals with some optimal control problems associated with the equations of steady-state, incompressible magnetohydrodynamics. These problems have direct applications to nuclear reactor technology, magnetic propulsion devices, and design of electromagnetic pumps. These problems are first put into an appropriate mathematical formulation. Then the existence of optimal solutions is proved. The use of Lagrange multiplier techniques is justified and an optimality system of equations is derived. The theory is applied to an example.
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    URL: http://dx.doi.org/10.1007/BF01185228
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  • 15
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    Generalized sentinels defined via least squares (1995)
    Springer
    Applied mathematics & optimization 31 (1995), S. 189-218 
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    ISSN: 1432-0606
    Keywords: Least squares ; Sentinels ; Optimal control ; Regularization ; Duality ; 93A30 ; 49K27
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We address the problem of monitoring a linear functional (c, x)Eof an unknown vectorx of a Hilbert spaceE, the available data being the observationz, in a Hilbert spaceF, of a vectorAx depending linearly onx through some known operatorAεℒ(E; F). WhenE=E 1×E 2,c=(c 1 0), andA is injective and defined through the solution of a partial differential equation, Lions ([6]–[8]) introduced sentinelssεF such that (s, Ax)Fis sensitive to x1 εE 1 but insensitive to x2 ε E2. In this paper we prove the existence, in the general case, of (i) a generalized sentinel (s, σ) ε ℱ ×E, where ℱ ⊃F withF dense in 80, such that for anya priori guess x0 ofx, we have 〈s, Ax〉ℱℱ + (σ, x0)E=(c, x)E, where x is the least-squares estimate ofx closest tox 0, and (ii) a family of regularized sentinels (s n , σ n ) ε F×E which converge to (s, σ). Generalized sentinels unify the least-squares approach (by construction !) and the sentinel approach (whenA is injective), and provide a general framework for the construction of “sentinels with special sensitivity” in the sense of Lions [8]).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01182788
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  • 16
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    An SQP method for optimal control of weakly singular Hammerstein integral equations (1996)
    Springer
    Applied mathematics & optimization 33 (1996), S. 227-252 
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    ISSN: 1432-0606
    Keywords: Lagrange-Newton method ; Sequential quadratic programming ; Optimal control ; Weakly singular Hammerstein integral equations ; Control constraints ; 49M05 ; 49K22
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We investigate local convergence of an SQP method for nonlinear optimal control of weakly singular Hammerstein integral equations. Sufficient conditions for local quadratic convergence of the method are discussed.
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    URL: http://dx.doi.org/10.1007/BF01204703
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  • 17
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    Optimal control of the chemotherapy of HIV (1997)
    Springer
    Journal of mathematical biology 35 (1997), S. 775-792 
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    ISSN: 1432-1416
    Keywords: Key words: Chemotherapy ; HIV ; Optimal control ; Ordinary differential equation system
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract.  Using an existing ordinary differential equation model which describes the interaction of the immune system with the human immunodeficiency virus (HIV), we introduce chemotherapy in an early treatment setting through a dynamic treatment and then solve for an optimal chemotherapy strategy. The control represents the percentage of effect the chemotherapy has on the viral production. Using an objective function based on a combination of maximizing benefit based on T cell counts and minimizing the systemic cost of chemotherapy (based on high drug dose/strength), we solve for the optimal control in the optimality system composed of four ordinary differential equations and four adjoint ordinary differential equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002850050076
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  • 18
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    Metric unidimensional scaling and global optimization (1996)
    Springer
    Journal of classification 13 (1996), S. 3-18 
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    ISSN: 1432-1343
    Keywords: Unidimensional scaling ; Seriation ; Local minima ; Global optimization ; Smoothing technique ; Multidimensional scaling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For the problem of metric unidimensional scaling, the number of local minima is estimated. For locating the globally optimal solution we develop an approach, called the “smoothing technique.” Although not guaranteed inevitably to locate the global optimum, the smoothing technique did so in all computational experiments where the global optimum was known.
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    URL: http://dx.doi.org/10.1007/BF01202579
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  • 19
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    Optimal control for a tandem network of queues with blocking (1997)
    Springer
    Acta mathematicae applicatae sinica 13 (1997), S. 425-437 
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    ISSN: 1618-3932
    Keywords: Optimal control ; tandem queueing system
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider a two-station tandem queue with no intermediate buffer. Jobs at the first station may be blocked when the following station is occupied by another job. The objective is to control the arrival and departure processes, subject to some capacity limits, so that an expected discounted profit function is maximized. We prove that the optimal control policy is of a threshold type, and the characterization of the threshold is provided.
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    URL: http://dx.doi.org/10.1007/BF02009552
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  • 20
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    Neural network controller for a multivariable model of submarine dynamics (1998)
    Springer
    Neural computing & applications 7 (1998), S. 295-308 
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    ISSN: 1433-3058
    Keywords: Adaptive ; Backpropagation ; Multivariable ; Neural networks ; Optimal control ; Submarine
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Recently, there have been many attempts to use neural networks as a feedback controller. However, most of the reported cases seek to control Single-Input Single-Output (SISO) systems using some sort of adaptive strategy. In this paper, we demonstrate that neural networks can be used for the control of complex multivariable, rather than simply SISO, systems. A modified direct control scheme using a neural network architecture is used with backpropagation as the adaptive algorithm. The proposed algorithm is designed for Multi-Input Multi-Output (MIMO) systems, and is similar to that proposed by Saerens and Soquet [1] and Goldenthal and Farrell [2] for (SISO) systems, and differs only in the form of the gradient approximation. As an example of the application of this approach, we investigate the control of the dynamics of a submarine vehicle with four inputs and four outputs, in which the differential stern, bow and rudder control surfaces are dynamically coordinated to cause the submarine to follow commanded changes in roll, yaw rate, depth rate and pitch attitude. Results obtained using this scheme are compared with those obtained using optimal linear quadratic control.
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    URL: http://dx.doi.org/10.1007/BF01428121
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  • 21
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    The linear quadratic optimal control problem for linear descriptor systems with variable coefficients (1997)
    Springer
    Mathematics of control, signals, and systems 10 (1997), S. 247-264 
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    ISSN: 1435-568X
    Keywords: Descriptor systems ; Differential algebraic equations ; Optimal control ; Strangeness index ; Riccati differential-algebraic equations ; Euler-Lagrange equations ; Linear feedback
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology , Mathematics , Technology
    Notes: Abstract We study linear quadratic optimal control problems for linear variable coefficient descriptor systems. Generalization from the case of standard control problems leads to several difficulties. We discuss a behavioral approach that solves some of these difficulties. Furthermore, an analysis of general rectangular systems is given and generalized Euler-Lagrange equations and Riccati differential algebraic equations are discussed.
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    URL: http://dx.doi.org/10.1007/BF01211506
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    Disturbance rejecting optimal regulation of hyperbolic systems (1995)
    Springer
    Mathematics of control, signals, and systems 8 (1995), S. 241-256 
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    ISSN: 1435-568X
    Keywords: Robust control ; H ∞ control ; Disturbance rejection ; Hyperbolic systems ; Optimal control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology , Mathematics , Technology
    Notes: Abstract Optimal regulation of hyperbolic systems in the presence of unknown exogenous and initial disturbances is considered. Necessary conditions for determining the optimal control that tracks a desired trajectory in the presence of disturbances are developed. These necessary conditions have the form of a twopoint boundary-value problem along with certain equality and inequality conditions. The results also characterize the worst possible disturbances that are accommodated by the optimum controller without any serious performance degradation. Numerical results on the control of a vibrating beam are presented.
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    URL: http://dx.doi.org/10.1007/BF01211861
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  • 23
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    Optimal Control of Linear Time-Varying Systems via Haar Wavelets (1999)
    Springer
    Journal of optimization theory and applications 103 (1999), S. 641-655 
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    ISSN: 1573-2878
    Keywords: Optimal control ; Haar wavelets
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper introduces the application of Haar wavelets to the optimal control synthesis for linear time-varying systems. Based upon some useful properties of Haar wavelets, a special product matrix, a related coefficient matrix, and an operational matrix of backward integration are proposed to solve the adjoint equation of optimization. The results obtained by the proposed Haar approach are almost the same as those obtained by the conventional Riccati method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1021740209084
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  • 24
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    Boundaries of the receding horizon control for interconnected systems (1995)
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    Journal of optimization theory and applications 84 (1995), S. 251-271 
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    ISSN: 1573-2878
    Keywords: Optimal control ; decentralized control ; interconnected systems ; receding horizon control ; rolling horizon control ; predictive control ; quadratic optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In recent years, the finite-horizon quadratic minimization problem has become popular in process control, where the horizon is constantly rolled back. In this paper, this type of control, which is also called the receding horizon control, is considered for interconnected systems. First, the receding horizon control equations are formulated; then, some stability conditions depending on the interconnection norms and the horizon lengths are presented. For ∈-coupled systems, stability results similar to centralized systems are obtained. For interconnected systems which are not ∈-coupled, the existence of a horizon length and a corresponding stabilizing receding horizon control are derived. Finally, the performance of a locally computed receding horizon control for time-invariant and time-varying systems with different updating intervals is examined in an example.
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    Deletion-by-infeasibility rule for DC-constrained global optimization (1995)
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    Journal of optimization theory and applications 84 (1995), S. 443-455 
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    ISSN: 1573-2878
    Keywords: Global optimization ; nonconvex programming ; mathematical programming ; DC-programming ; branch-and-bound methods
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    Topics: Mathematics
    Notes: Abstract In Ref. 1, a general class of branch-and-bound methods was proposed by Horst for solving global optimization problems. One of the main contributions of Ref. 1 was the opportunity of handling partition elements whose feasibility is not known. Deletion-by-infeasibility rules were presented for problems where the feasible set is convex, is defined by finitely many convex and reverse convex constraints, or is defined by Lipschitzian inequalities. In this note, we propose a new deletion-by-infeasibility rule for problems whose feasible set is defined by functions representable as differences of convex functions.
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    Time-optimal extension and retraction of robots: Numerical analysis of the switching structure (1995)
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    Journal of optimization theory and applications 84 (1995), S. 589-616 
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    ISSN: 1573-2878
    Keywords: Optimal control ; robotics ; path planning ; maximum principle ; multi-point boundary value problems ; multiple shooting
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    Topics: Mathematics
    Notes: Abstract The problem of the time-optimal control of robot manipulators is of importance because of its potential for increasing the productivity of assembly lines. This work is part of a series of papers by the authors on this topic using direct and indirect methods of optimization. A cylindrical robot or a spherical polar robot constrained to the horizontal plane is considered, and optimal solutions for radial maneuvers are generated. Indirect methods are employed in order to establish the switching structure of the solutions. The results show that even such apparently simple maneuvers as extension or retraction of a robot with a prismatic joint can produce very complex optimal solutions. Time-optimal retraction can exhibit ten different switching structures with eight switching points and two singular arcs.
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    Second-order sufficient conditions for control problems with mixed control-state constraints (1995)
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    Journal of optimization theory and applications 86 (1995), S. 649-667 
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    ISSN: 1573-2878
    Keywords: Optimal control ; mixed control-state constraints ; Hamilton-Jacobi inequality ; second-order sufficient conditions ; parametric optimization ; Riccati equations
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    Topics: Mathematics
    Notes: Abstract References 1–4 develop second-order sufficient conditions for local minima of optimal control problems with state and control constraints. These second-order conditions tighten the gap between necessary and sufficient conditions by evaluating a positive-definiteness criterion on the tangent space of the active constraints. The purpose of this paper is twofold. First, we extend the methods in Refs. 3, 4 and include general boundary conditions. Then, we relate the approach to the two-norm approach developed in Ref. 5. A direct sufficiency criterion is based on a quadratic function that satisfies a Hamilton-Jacobi inequality. A specific form of such a function is obtained by applying the second-order sufficient conditions to a parametric optimization problem. The resulting second-order positive-definiteness conditions can be verified by solving Riccati equations.
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    Decomposition approach for the global minimization of biconcave functions over polytopes (1996)
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    Journal of optimization theory and applications 88 (1996), S. 561-583 
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    ISSN: 1573-2878
    Keywords: Global optimization ; biconcave programming ; concave minimization ; bilinear and quadratic programming ; branch-and-bound algorithms ; outer approximations.
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    Notes: Abstract A decomposition approach is proposed for minimizing biconcave functions over polytopes. Important special cases include concave minimization, bilinear and indefinite quadratic programming for which new algorithms result. The approach introduces a new polyhedral partition and combines branch-and-bound techniques, outer approximation, and projection of polytopes in a suitable way.
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    Generalized primal-relaxed dual approach for global optimization (1996)
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    Journal of optimization theory and applications 90 (1996), S. 417-434 
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    ISSN: 1573-2878
    Keywords: Global optimization ; primal-relaxed dual approach ; penalty methods ; nonsmooth optimization
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    Notes: Abstract A generalized primal-relaxed dual algorithm for global optimization is proposed and its convergence is proved. The (GOP) algorithm of Floudas and Visweswaran (Refs. 1–2) is shown to be a special case of this general algorithm. Within the proposed framework, the algorithm of Floudas and Visweswaran (Refs. 1–2) is further extended to the nonsmooth case. A penalty implementation of the extended (GOP) algorithm is studied to improve its efficiency.
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    Use of Augmented Lagrangian Methods for the Optimal Control of Obstacle Problems (1997)
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    Journal of optimization theory and applications 95 (1997), S. 101-126 
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    ISSN: 1573-2878
    Keywords: Optimal control ; Lagrange multipliers ; augmented Lagrangians ; variational inequalities
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    Topics: Mathematics
    Notes: Abstract We investigate optimal control problems governed by variational inequalities involving constraints on the control, and more precisely the example of the obstacle problem. In this paper, we discuss some augmented Lagrangian algorithms to compute the solution.
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    Optimal Control of a Chemical Vapor Deposition Reactor (1998)
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    Journal of optimization theory and applications 97 (1998), S. 623-644 
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    ISSN: 1573-2878
    Keywords: Optimal control ; bang-bang control ; free boundary problems ; parabolic equations ; homogenizations ; optimality conditions
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    Notes: Abstract We study a simple model of chemical vapor deposition on a silicon wafer. The control is the flux of chemical species, and the objective is to grow the semiconductor film so that its surface attains a prescribed profile as nearly as possible. The surface is spatially fast oscillating due to the small feature scale, and therefore the problem is formulated in terms of its homogenized approximation. We prove that the optimal control is bang-bang, and we use this information to develop a numerical scheme for computing the optimal control.
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    Stochastic Methods for Practical Global Optimization (1998)
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    Journal of global optimization 13 (1998), S. 433-444 
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    ISSN: 1573-2916
    Keywords: Adaptive search ; Global optimization ; Multi-disciplinary optimization ; Simulated annealing
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    Topics: Mathematics
    Notes: Abstract Engineering design problems often involve global optimization of functions that are supplied as ‘black box’ functions. These functions may be nonconvex, nondifferentiable and even discontinuous. In addition, the decision variables may be a combination of discrete and continuous variables. The functions are usually computationally expensive, and may involve finite element methods. An engineering example of this type of problem is to minimize the weight of a structure, while limiting strain to be below a certain threshold. This type of global optimization problem is very difficult to solve, yet design engineers must find some solution to their problem – even if it is a suboptimal one. Sometimes the most difficult part of the problem is finding any feasible solution. Stochastic methods, including sequential random search and simulated annealing, are finding many applications to this type of practical global optimization problem. Improving Hit-and-Run (IHR) is a sequential random search method that has been successfully used in several engineering design applications, such as the optimal design of composite structures. A motivation to IHR is discussed as well as several enhancements. The enhancements include allowing both continuous and discrete variables in the problem formulation. This has many practical advantages, because design variables often involve a mixture of continuous and discrete values. IHR and several variations have been applied to the composites design problem. Some of this practical experience is discussed.
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    Two Strategies of Adaptive Cluster Covering with Descent and Their Comparison to Other Algorithms (1999)
    Springer
    Journal of global optimization 14 (1999), S. 55-78 
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    ISSN: 1573-2916
    Keywords: Adaptation ; Clustering ; Covering ; Descent ; Global optimization ; Parameter identification
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    Notes: Abstract Two strategies of randomized search, namely adaptive cluster covering (ACCO), and adaptive cluster covering with descent (ACD), are introduced and positioned in the group of the global optimization techniques. Several algorithms based on these new strategies are compared with other techniques of global randomized search in terms of effectiveness, efficiency and reliability. The other techniques include two versions of multistart, two versions of the controlled random search (CRS2 and CRS4) and the canonical genetic algorithm. Thirteen minimization problems including two parameter identification problems (for a flexible membrane mirror model and a hydrologic model) are solved. The algorithm ACCO, and a version of CRS4 algorithm (Ali and Storey 1994) show the highest efficiency, effectiveness and reliability. The second new algorithm, ACD, is in some runs very efficient and effective, but its reliability needs further improvement.
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    An adaptive simulated annealing algorithm for global optimization over continuous variables (1995)
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    Journal of global optimization 6 (1995), S. 1-37 
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    ISSN: 1573-2916
    Keywords: Global optimization ; simulated annealing ; Monte Carlo optimization
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    Notes: Abstract A method is presented for attempting global minimization for a function of continuous variables subject to constraints. The method, calledAdaptive Simulated Annealing (ASA), is distinguished by the fact that the fixed temperature schedules and step generation routines that characterize other implementations are here replaced by heuristic-based methods that effectively eliminate the dependence of the algorithm's overall performance on user-specified control parameters. A parallelprocessing version of ASA that gives increased efficiency is presented and applied to two standard problems for illustration and comparison.
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    Parallel methods for verified global optimization practice and theory (1996)
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    Journal of global optimization 9 (1996), S. 1-22 
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    Keywords: Global optimization ; parallel computing ; interval arithmetic ; branch and bound ; dynamic load balancing
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    Notes: Abstract We present a new parallel method for verified global optimization, using a centralized mediator for the dynamic load balancing. The new approach combines the advantages of two previous models, the master slave model and the processor farm. Numerical results show the efficiency of this new method. For a large number of problems at least linear speedup is reached. The efficiency of this new method is also confirmed by a comparison with other parallel methods for verified global optimization. A theoretical study proves that using the best-first strategy to choose the next box for subdivision, no real superlinear speedup may be expected concerning the number of iterations. Moreover, the potential of parallelization of methods of verified global optimization is discussed in general.
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    Second-order sufficient optimality conditions for local and global nonlinear programming (1996)
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    Journal of global optimization 9 (1996), S. 141-151 
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    Keywords: Global optimization ; optimality condition ; second-order sufficient condition ; verification of convexity ; interval analysis ; primary 90C30 ; secondary 65G10
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    Topics: Mathematics
    Notes: Abstract This paper presents a new approach to the sufficient conditions of nonlinear programming. Main result is a sufficient condition for the global optimality of a Kuhn-Tucker point. This condition can be verified constructively, using a novel convexity test based on interval analysis, and is guaranteed to prove global optimality of strong local minimizers for sufficiently narrow bounds. Hence it is expected to be a useful tool within branch and bound algorithms for global optimization.
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    Global and Local Quadratic Minimization (1997)
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    Journal of global optimization 10 (1997), S. 77-90 
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    ISSN: 1573-2916
    Keywords: Global optimization ; parametric quadratic programming ; non-convex quadratic program.
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    Notes: Abstract We present a method which when applied to certain non-convex QP will locatethe globalminimum, all isolated local minima and some of the non-isolated localminima. The method proceeds by formulating a (multi) parametric convex QP interms ofthe data of the given non-convex QP. Based on the solution of the parametricQP,an unconstrained minimization problem is formulated. This problem ispiece-wisequadratic. A key result is that the isolated local minimizers (including theglobalminimizer) of the original non-convex problem are in one-to-one correspondencewiththose of the derived unconstrained problem. The theory is illustrated with several numerical examples. A numericalprocedure isdeveloped for a special class of non-convex QP's. It is applied to a problemfrom theliterature and verifies a known global optimum and in addition, locates apreviously unknown local minimum.
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    Global Optimization of Nonconvex Polynomial Programming Problems Having Rational Exponents (1998)
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    Journal of global optimization 12 (1998), S. 267-283 
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    ISSN: 1573-2916
    Keywords: Polynomial programs ; Reformulation-Linearization Technique (RLT) ; Nonconvex programming ; Global optimization
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    Topics: Mathematics
    Notes: Abstract This paper considers the solution of nonconvex polynomial programming problems that arise in various engineering design, network distribution, and location-allocation contexts. These problems generally have nonconvex polynomial objective functions and constraints, involving terms of mixed-sign coefficients (as in signomial geometric programs) that have rational exponents on variables. For such problems, we develop an extension of the Reformulation-Linearization Technique (RLT) to generate linear programming relaxations that are embedded within a branch-and-bound algorithm. Suitable branching or partitioning strategies are designed for which convergence to a global optimal solution is established. The procedure is illustrated using a numerical example, and several possible extensions and algorithmic enhancements are discussed.
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    Tracking Demands in Optimal Control of Managerial Systems with Continuously-Divisible, Doubly Constrained Resources (1998)
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    Journal of global optimization 13 (1998), S. 43-59 
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    ISSN: 1573-2916
    Keywords: Resource constrained scheduling ; renewable and nonrenewable resources ; Optimal control
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    Notes: Abstract The paper addresses problems of allocating continuously divisible resources among multiple production activities. The resources are allowed to be doubly constrained, so that both usage at every point of time and cumulative consumption over a planning horizon are limited as it is often the case in project and production scheduling. The objective is to track changing in time demands for the activities as closely as possible. We propose a general continuous-time model that states the problem in a form of the optimal control problem with non-linear speed-resource usage functions. With the aid of the maximum principle, properties of the solutions are derived to characterize optimal resource usage policies. On the basis of this analytical investigation, numerical scheduling methods are suggested and computationally studied.
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    Global Optimization in Practice: An Application to Interactive Multiple Objective Linear Programming (1998)
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    Journal of global optimization 12 (1998), S. 353-372 
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    ISSN: 1573-2916
    Keywords: Multiple objective linear programming ; Optimization over the efficient set ; Interactive methods ; Global optimization ; Citrus rootstockselection
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    Topics: Mathematics
    Notes: Abstract A multiple objective linear programming problem (P) involves the simultaneous maximization of two or more conflicting linear objective functions over a nonempty polyhedron X. Many of the most popular methods for solving this type of problem, including many well-known interactive methods, involve searching the efficient set X E of the problem. Generally, however, X E is a complicated, nonconvex set. As a result, concepts and methods from global optimization may be useful in searching X E. In this paper, we will explain in theory, and show via an actual application to citrus rootstock selection in Florida, how the potential usefulness of the well-known interactive method STEM for solving problem (P) in this way, can depend crucially upon how accurately certain global optimization problems involving minimizations over X E are solved. In particular, we will show both in theory and in practice that the choice of whether to use the popular but unreliable ‘payoff table’ approach or to use one of the lesser known, more accurate global optimization methods to solve these problems can determine whether STEM succeeds or fails as a decision aid. Several lessons and conclusions of transferable value derived from this research are also given.
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    A Generic Global Optimization Algorithm for the Chemical and Phase Equilibrium Problem (1998)
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    Journal of global optimization 12 (1998), S. 325-351 
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    Keywords: Chemical and phase equilibrium ; convexity ; Gibbs free energy ; Global optimization ; Non-convex optimization ; Tangent-plane criterion
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    Notes: Abstract This paper addresses the problem of finding the number, K, of phases present at equilibrium and their composition, in a chemical mixture of n s substances. This corresponds to the global minimum of the Gibbs free energy of the system, subject to constraints representing m b independent conserved quantities, where m b=n s when no reaction is possible and m b ≤ n e +1 when reaction is possible and n e is the number of elements present. After surveying previous work in the field and pointing out the main issues, we extend the necessary and sufficient condition for global optimality based on the ‘reaction tangent-plane criterion’, to the case involving different thermodynamical models (multiple phase classes). We then present an algorithmic approach that reduces this global optimization problem (involving a search space of m b(n s-1) dimensions) to a finite sequence of local optimization steps inK(n s-1) -space, K ≤ m b, and global optimization steps in (n s-1)-space. The global step uses the tangent-plane criterion to determine whether the current solution is optimal, and, if it is not, it finds an improved feasible solution either with the same number of phases or with one added phase. The global step also determines what class of phase (e.g. liquid or vapour) is to be added, if any phase is to be added. Given a local minimization procedure returning a Kuhn–Tucker point and a global optimization procedure (for a lower-dimensional search space) returning a global minimum, the algorithm is proved to converge to a global minimum in a finite number of the above local and global steps. The theory is supported by encouraging computational results.
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    Optimal control in light traffic Markov decision processes (1997)
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    Mathematical methods of operations research 45 (1997), S. 63-79 
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    ISSN: 1432-5217
    Keywords: Optimal control ; Markov decision processes ; light traffic ; discounted cost ; average cost
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    Topics: Mathematics , Economics
    Notes: Abstract We consider Markov Decision Processes under light traffic conditions. We develop an algorithm to obtain asymptotically optimal policies for both the total discounted and the average cost criterion. This gives a general framework for several light traffic results in the literature. We illustrate the method by deriving the asymptotically optimal control of a simple ATM network.
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    An existence and uniqueness theorem for one optimal control problem (1995)
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    Periodica mathematica Hungarica 30 (1995), S. 61-65 
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    ISSN: 1588-2829
    Keywords: Primary 49J20 ; Secondary 65L10 ; Optimal control ; boundary value problem ; existence theory
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    Topics: Mathematics
    Notes: Abstract In this paper we investigate the existence and uniqueness for an optimal control problem with processes described by a quasilinear parabolic equation with controls in coefficients and the right side of this equation.
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    Feasible Direction Algorithm for Optimal Control Problems with State and Control Constraints: Implementation (1999)
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    Journal of optimization theory and applications 101 (1999), S. 623-649 
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    ISSN: 1573-2878
    Keywords: Optimal control ; state-constrained problems ; numerical algorithms
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    Notes: Abstract In this paper, we describe the implementation aspects of an optimization algorithm for optimal control problems with control, state, and terminal constraints presented in our earlier paper. The important aspect of the implementation is that, in the direction-finding subproblems, it is necessary only to impose the state constraint at relatively few points in the time involved. This contributes significantly to the algorithmic efficiency. The algorithm is applied to solve several optimal control problems, including the problem of the abort landing of an aircraft in the presence of windshear.
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    Global State Space Approach for the Efficient Numerical Solution of State-Constrained Trajectory Optimization Problems (1999)
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    Journal of optimization theory and applications 103 (1999), S. 75-93 
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    ISSN: 1573-2878
    Keywords: Optimal control ; state inequality constraints ; global state space form ; differential-algebraic boundary-value problems ; minimal coordinates ; trajectory optimization
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    Topics: Mathematics
    Notes: Abstract A new approach based on a global state space form is introduced for solving trajectory optimization problems with state inequality constraints via indirect methods. The use of minimal coordinates on a boundary arc of the state constraint eliminates severe problems, which occur for standard methods and are due to the appearance of differential-algebraic boundary-value problems. Together with a hybrid approach and a careful treatment of some interior-point conditions, we obtain an efficient and reliable solution method.
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    State Analysis and Optimal Control of Time-Varying Discrete Systems via Haar Wavelets (1999)
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    Journal of optimization theory and applications 103 (1999), S. 623-640 
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    ISSN: 1573-2878
    Keywords: Optimal control ; discrete systems ; Haar wavelets
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    Topics: Mathematics
    Notes: Abstract The state analysis and optimal control of time-varying discrete systems via Haar wavelets are the main tasks of this paper. First, we introduce the definition of discrete Haar wavelets. Then, a comparison between Haar wavelets and other orthogonal functions is given. Based upon some useful properties of the Haar wavelets, a special product matrix and a related coefficient matrix are proposed; also, a shift matrix and a summation matrix are derived. These matrices are very effective in solving our problems. The local property of the Haar wavelets is applied to shorten the calculation procedures.
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    Direct training method for a continuous-time nonlinear optimal feedback controller (1995)
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    Journal of optimization theory and applications 84 (1995), S. 509-528 
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    ISSN: 1573-2878
    Keywords: Optimal control ; nonlinear feedback control ; feedforward neural networks
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    Topics: Mathematics
    Notes: Abstract The solutions of most nonlinear optimal control problems are given in the form of open-loop optimal control which is computed from a given fixed initial condition. Optimal feedback control can in principle be obtained by solving the corresponding Hamilton-Jacobi-Bellman dynamic programming equation, though in general this is a difficult task. We propose a practical and effective alternative for constructing an approximate optimal feedback controller in the form of a feedforward neural network, and we justify this choice by several reasons. The controller is capable of approximately minimizing an arbitrary performance index for a nonlinear dynamical system for initial conditions arising from a nontrivial bounded subset of the state space. A direct training algorithm is proposed and several illustrative examples are given.
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    New LP bound in multivariate Lipschitz optimization: Theory and applications (1995)
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    Journal of optimization theory and applications 86 (1995), S. 369-388 
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    ISSN: 1573-2878
    Keywords: Global optimization ; Lipschitz optimization ; systems of inequalities ; branch-and-bounds methods
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    Topics: Mathematics
    Notes: Abstract Most numerically promising methods for solving multivariate unconstrained Lipschitz optimization problems of dimension greater than two use rectangular or simplicial branch-and-bound techniques with computationally cheap but rather crude lower bounds. Generalizations to constrained problems, however, require additional devices to detect sufficiently many infeasible partition sets. In this article, a new lower bounding procedure is proposed for simplicial methods yielding considerably better bounds at the expense of two linear programs in each iteration. Moreover, the resulting approach can solve easily linearly constrained problems, since in this case infeasible partition sets are automatically detected by the lower bounding procedure. Finally, it is shown that the lower bounds can be further improved when the method is applied to solve systems of inequalities. Implementation issues, numerical experiments, and comparisons are discussed in some detail.
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    Marketable permits in a stochastic dynamic model of the firm (1996)
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    Journal of optimization theory and applications 89 (1996), S. 129-155 
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    ISSN: 1573-2878
    Keywords: Optimal control ; stochastic control ; Hamilton-Jacobi-Bellman equation ; environmental problems ; dynamics of the firm ; marketable permits
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    Topics: Mathematics
    Notes: Abstract This contribution attempts to determine the effects of environmental regulation on the growth of an individual firm. Here, it is assumed that the firm revenue is stochastic. The government tries to reduce pollution by creating a market on which the firm has to buy permits in order to be allowed to pollute the environment. Pollution is an inevitable byproduct of the firm production process, and in our model the firm is offered two ways to deal with it. The first is to buy marketable permits, and the second is to clean up pollution which can be achieved through investing in abatement capital stock. It turns out that the firm optimal trajectory consists of at most seven different policies. They can be depicted in a feedback diagram from which we can conclude that, provided that the firm never faces a shortage of cash, productive and abatement capital stocks ultimately reach their equilibrium levels where marginal revenue equals marginal costs.
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    Linearly constrained global minimization of functions with concave minorants (1996)
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    Journal of optimization theory and applications 88 (1996), S. 751-763 
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    Keywords: Global optimization ; functions with concave minorants ; d.c. optimization ; branch-and-bounds methods ; systems of inequalities
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    Notes: Abstract In this note, we show how a recent approach for solving linearly constrained multivariate Lipschitz optimization problems and corresponding systems of inequalities can be generalized to solve optimization problems where the objective function is only assumed to possess a concave minorant at each point. This class of functions includes not only Lipschitz functions and some generalizations, such as certain ρ-convex functions and Hölder functions with exponent greater than one, but also all functions which can be expressed as differences of two convex functions (d.c. functions). Thus, in particular, a new approach is obtained for the important problem of minimizing a d.c. function over a polytope.
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    Homogenization of Optimal Control Problems for Functional Differential Equations (1997)
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    Journal of optimization theory and applications 93 (1997), S. 103-119 
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    ISSN: 1573-2878
    Keywords: Optimal control ; variational convergence ; functional differential equations
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    Notes: Abstract The paper deals with the variational convergence of a sequence of optimal control problems for functional differential state equations with deviating argument. Variational limit problems are found under various conditions of convergence of the input data. It is shown that, upon sufficiently weak assumptions on convergence of the argument deviations, the limit problem can assume a form different from that of the whole sequence. In particular, it can be either an optimal control problem for an integro-differential equation or a purely variational problem. Conditions are found under which the limit problem preserves the form of the original sequence.
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    New Algorithm for Computing LQ Suboptimal Output Feedback Gains of Decentralized Control Systems (1997)
    Springer
    Journal of optimization theory and applications 93 (1997), S. 597-607 
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    ISSN: 1573-2878
    Keywords: Optimal control ; gradient flows ; optimization ; decentralized systems ; performance index
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    Notes: Abstract In this paper, the problem of computing the suboptimal output feedback gains of decentralized control systems is investigated. First, the problem is formulated. Then, the gradient matrices based on the index function are derived and a new algorithm is established based on some nice properties. This algorithm shows that a suboptimal gain can be computed by solving several ordinary differential equations (ODEs). In order to find an initial condition for the ODEs, an algorithm for finding a stabilizing output feedback gain is exploited, and the convergence of this algorithm is discussed. Finally, an example is given to illustrate the proposed algorithm.
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    Lagrange Duality and Partitioning Techniques in Nonconvex Global Optimization (1997)
    Springer
    Journal of optimization theory and applications 95 (1997), S. 347-369 
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    ISSN: 1573-2878
    Keywords: Global optimization ; duality gap ; branch-and-bound techniques ; partly convex programs ; bilinear constraints ; sum of ratios ; reverse convex constraints
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    Notes: Abstract It is shown that, for very general classes of nonconvex global optimization problems, the duality gap obtained by solving a corresponding Lagrangian dual in reduced to zero in the limit when combined with suitably refined partitioning of the feasible set. A similar result holds for partly convex problems where exhaustive partitioning is applied only in the space of nonconvex variables. Applications include branch-and-bound approaches for linearly constrained problems where convex envelopes can be computed, certain generalized bilinear problems, linearly constrained optimization of the sum of ratios of affine functions, and concave minimization under reverse convex constraints.
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    Global Optimization Requires Global Information (1998)
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    Journal of optimization theory and applications 96 (1998), S. 575-588 
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    ISSN: 1573-2878
    Keywords: Global optimization ; convergence ; stochastic algorithms ; deterministic algorithms
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    Notes: Abstract There are many global optimization algorithms which do not use global information. We broaden previous results, showing limitations on such algorithms, even if allowed to run forever. We show that deterministic algorithms must sample a dense set to find the global optimum value and can never be guaranteed to converge only to global optimizers. Further, analogous results show that introducing a stochastic element does not overcome these limitations. An example is simulated annealing in practice. Our results show that there are functions for which the probability of success is arbitrarily small.
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    On a Class of Optimal Control Problems with State Jumps (1998)
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    Journal of optimization theory and applications 98 (1998), S. 65-82 
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    ISSN: 1573-2878
    Keywords: Optimal control ; switching times ; state jumps ; transformations ; optimal parameter selection problem ; automatic differentiation
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    Notes: Abstract In this paper, we consider a class of optimal control problems in which the dynamical system involves a finite number of switching times together with a state jump at each of these switching times. The locations of these switching times and a parameter vector representing the state jumps are taken as decision variables. We show that this class of optimal control problems is equivalent to a special class of optimal parameter selection problems. Gradient formulas for the cost functional and the constraint functional are derived. On this basis, a computational algorithm is proposed. For illustration, a numerical example is included.
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    Global Interval Methods for Local Nonsmooth Optimization (1999)
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    Journal of global optimization 14 (1999), S. 157-179 
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    ISSN: 1573-2916
    Keywords: Global optimization ; Interval methods ; Local nonsmooth optimization ; Scientific computing
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    Notes: Abstract An interval method for determining local solutions of nonsmooth unconstrained optimization problems is discussed. The objective function is assumed to be locally Lipschitz and to have appropriate interval inclusions. The method consists of two parts, a local search and a global continuation and termination. The local search consists of a globally convergent descent algorithm showing similarities to ε-bundle methods. While ε-bundle methods use polytopes as inner approximations of the ε-subdifferentials, which are the main tools of almost all bundle concepts, our method uses axes parallel boxes as outer approximations of the ε-subdifferentials. The boxes are determined almost automatically with inclusion techniques of interval arithmetic. The dimension of the boxes is equal to the dimension of the problem and remains constant during the whole computation. The application of boxes does not suffer from the necessity to invest methodical and computational efforts to adapt the polytopes to the latest state of the computation as well as to simplify them when the number of vertices becomes too large, as is the case with the polytopes. The second part of the method applies interval techniques of global optimization to the approximative local solution obtained from the search of the first part in order to determine guaranteed error bounds or to improve the solution if necessary. We present prototype algorithms for both parts of the method as well as a complete convergence theory for them and demonstrate how outer approximations can be obtained.
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    Extension of Piyavskii's Algorithm to Continuous Global Optimization (1999)
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    Journal of global optimization 14 (1999), S. 205-216 
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    Keywords: Global optimization ; Lipschitz continuity ; Piyavskii's algorithm ; Uniform Continuity
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    Notes: Abstract We use the simple, but little-known, result that a uniformly continuous function on a convex set is ε-Lipschitz (as defined below) to extend Piyavskii's algorithm for Lipschitz global optimization to the larger domain of continuous (not-necessarily-Lipschitz) global optimization.
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    A Nonsmooth Global Optimization Technique Using Slopes: The One-Dimensional Case (1999)
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    Journal of global optimization 14 (1999), S. 365-393 
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    ISSN: 1573-2916
    Keywords: Branch-and-bound ; Global optimization ; Interval arithmetic ; Interval slopes
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    Notes: Abstract In this paper we introduce a pruning technique based on slopes in the context of interval branch-and-bound methods for nonsmooth global optimization. We develop the theory for a slope pruning step which can be utilized as an accelerating device similar to the monotonicity test frequently used in interval methods for smooth problems. This pruning step offers the possibility to cut away a large part of the box currently investigated by the optimization algorithm. We underline the new technique's efficiency by comparing two variants of a global optimization model algorithm: one equipped with the monotonicity test and one equipped with the pruning step. For this reason, we compared the required CPU time, the number of function and derivative or slope evaluations, and the necessary storage space when solving several smooth global optimization problems with the two variants. The paper concludes on the test results for several nonsmooth examples.
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    A New Semidefinite Programming Bound for Indefinite Quadratic Forms Over a Simplex (1999)
    Springer
    Journal of global optimization 14 (1999), S. 357-364 
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    ISSN: 1573-2916
    Keywords: Global optimization ; Nonconvex quadratic programming ; Semidefinite programming
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    Notes: Abstract The paper describes a method for computing a lower bound of the global minimum of an indefinite quadratic form over a simplex. The bound is derived by computing an underestimator of the convex envelope by solving a semidefinite program (SDP). This results in a convex quadratic program (QP). It is shown that the optimal value of the QP is a lower bound of the optimal value of the original problem. Since there exist fast (polynomial time) algorithms for solving SDP's and QP's the bound can be computed in reasonable time. Numerical experiments indicate that the relative error of the bound is about 10 percent for problems up to 20 variables, which is much better than a known SDP bound.
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    Stochastic Global Optimization: Problem Classes and Solution Techniques (1999)
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    Journal of global optimization 14 (1999), S. 437-447 
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    Keywords: Global optimization ; Problem features ; Problem classes ; Test problems ; Solution techniques
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    Notes: Abstract There is a lack of a representative set of test problems for comparing global optimization methods. To remedy this a classification of essentially unconstrained global optimization problems into unimodal, easy, moderately difficult, and difficult problems is proposed. The problem features giving this classification are the chance to miss the region of attraction of the global minimum, embeddedness of the global minimum, and the number of minimizers. The classification of some often used test problems are given and it is recognized that most of them are easy and some even unimodal. Global optimization solution techniques treated are global, local, and adaptive search and their use for tackling different classes of problems is discussed. The problem of fair comparison of methods is then adressed. Further possible components of a general global optimization tool based on the problem classes and solution techniques is presented.
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    Global optimization of arborescent multilevel inventory systems (1995)
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    Journal of global optimization 6 (1995), S. 269-292 
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    ISSN: 1573-2916
    Keywords: Global optimization ; inventory problems ; discrete Hamilton-Jacobi-Bellman equations ; quasi-variational inequalities ; subsolutions and supersolutions
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    Notes: Abstract We consider the numerical resolution of hierarchical inventory problems under global optimization. First we describe the model as well as the dynamical stochastic system and the impulse controls involved. Next we characterize the optimal cost function and we formulate the Hamilton-Jacobi-Bellman equations. We present a numerical scheme and a fast algorithm of resolution, with a result on the speed of convergence. Finally, we apply the discretization method to some examples where we show the usefulness of the proposed numerical method as well as the advantages of operating under global optimization.
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    A branch and bound algorithm for bound constrained optimization problems without derivatives (1995)
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    Journal of global optimization 7 (1995), S. 297-331 
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    ISSN: 1573-2916
    Keywords: 49D37 ; 65G10 ; Global optimization ; interval arithmetic
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    Topics: Mathematics
    Notes: Abstract In this paper, we give a new branch and bound algorithm for the global optimization problem with bound constraints. The algorithm is based on the use of inclusion functions. The bounds calculated for the global minimum value are proved to be correct, all rounding errors are rigorously estimated. Our scheme attempts to exclude most “uninteresting” parts of the search domain and concentrates on its “promising” subsets. This is done as fast as possible (by involving local descent methods), and uses little information as possible (no derivatives are required). Numerical results for many well-known problems as well as some comparisons with other methods are given.
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    Simple global minimization algorithm for one-variable rational functions (1995)
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    Journal of global optimization 6 (1995), S. 293-311 
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    Keywords: Global optimization ; univariate optimization ; polynomials ; rational functions ; Storm's chain
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    Notes: Abstract Sturm's chain technique for evaluation of a number of real roots of polynomials is applied to construct a simple algorithm for global optimization of polynomials or generally for rational functions of finite global minimal value. The method can be applied both to find the global minimum in an interval or without any constraints. It is shown how to use the method to minimize globally a truncated Fourier series. The results of numerical tests are presented and discussed. The cost of the method scales as the square of the degree of the polynomial.
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    On the selection of subdivision directions in interval branch-and-bound methods for global optimization (1995)
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    Journal of global optimization 7 (1995), S. 183-207 
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    ISSN: 1573-2916
    Keywords: Global optimization ; interval arithmetic ; branch-and-bound ; interval subdivision
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    Notes: Abstract This paper investigates the influence of the interval subdivision selection rule on the convergence of interval branch-and-bound algorithms for global optimization. For the class of rules that allows convergence, we study the effects of the rules on a model algorithm with special list ordering. Four different rules are investigated in theory and in practice. A wide spectrum of test problems is used for numerical tests indicating that there are substantial differences between the rules with respect to the required CPU time, the number of function and derivative evaluations, and the necessary storage space. Two rules can provide considerable improvements in efficiency for our model algorithm.
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    Global optimization of Hölder functions (1996)
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    Journal of global optimization 8 (1996), S. 323-348 
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    Keywords: Global optimization ; Hölder functions ; Lipschitz optimization
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    Notes: Abstract We propose a branch-and-bound framework for the global optimization of unconstrained Hölder functions. The general framework is used to derive two algorithms. The first one is a generalization of Piyavskii's algorithm for univariate Lipschitz functions. The second algorithm, using a piecewise constant upper-bounding function, is designed for multivariate Hölder functions. A proof of convergence is provided for both algorithms. Computational experience is reported on several test functions from the literature.
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    An algorithm for maximizing a convex function over a simple set (1996)
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    Journal of global optimization 8 (1996), S. 379-391 
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    Keywords: Global optimization ; optimality condition ; convex function ; numerical algorithm ; simple set
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    Topics: Mathematics
    Notes: Abstract The problem of maximizing a convex function on a so-called simple set is considered. Based on the optimality conditions [19], an algorithm for solving the problem is proposed. This numerical algorithm is shown to be convergent. The proposed algorithm has been implemented and tested on a variety of test problems.
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    A branch-and-reduce approach to global optimization (1996)
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    Journal of global optimization 8 (1996), S. 107-138 
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    Keywords: Global optimization ; range reduction ; branch-and-bound ; polynomial programming ; multiplicative programming ; mixed-integer nonlinear programming ; quadratic programming
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    Notes: Abstract This paper presents valid inequalities and range contraction techniques that can be used to reduce the size of the search space of global optimization problems. To demonstrate the algorithmic usefulness of these techniques, we incorporate them within the branch-and-bound framework. This results in a branch-and-reduce global optimization algorithm. A detailed discussion of the algorithm components and theoretical properties are provided. Specialized algorithms for polynomial and multiplicative programs are developed. Extensive computational results are presented for engineering design problems, standard global optimization test problems, univariate polynomial programs, linear multiplicative programs, mixed-integer nonlinear programs and concave quadratic programs. For the problems solved, the computer implementation of the proposed algorithm provides very accurate solutions in modest computational time.
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    A Global Optimization Algorithm for Multivariate Functions with Lipschitzian First Derivatives (1997)
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    Journal of global optimization 10 (1997), S. 257-281 
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    ISSN: 1573-2916
    Keywords: Global optimization ; multiextremal algorithms ; Lipschitzian first derivatives ; convergence ; numerical experiments.
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    Notes: Abstract In this paper we propose a new multi-dimensional methodto solve unconstrained global optimization problems with Lipschitzianfirst derivatives. The method is based on apartition scheme that subdivides the search domain into a set of hypercubesin the course of optimization. This partitioning is regulated by thedecision rule that provides evaluation of the "importance"of each generated hypercube and selection of some partition element to performthe next iteration. Sufficient conditions of global convergence for the newmethod are investigated. Results of numerical experiments are alsopresented.
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    Optimal Control for an Obstruction Problem (1999)
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    Journal of optimization theory and applications 100 (1999), S. 435-439 
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    Keywords: Optimal control ; flow round obstacles ; Pontryagin principle ; discretization
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    Notes: Abstract A question of flow around an obstacle leads to an optimal control problem. If an optimum path exists, then it is calculable from the Pontryagin principle. The optimum is verified to be reached, using a discretization of the problem.
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    Dynamic Programming Method for Constrained Discrete-Time Optimal Control (1999)
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    Journal of optimization theory and applications 101 (1999), S. 259-283 
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    ISSN: 1573-2878
    Keywords: Optimal control ; dynamic programming ; interior-point methods ; sequential quadratic programming
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    Notes: Abstract A dynamic programming method is presented for solving constrained, discrete-time, optimal control problems. The method is based on an efficient algorithm for solving the subproblems of sequential quadratic programming. By using an interior-point method to accommodate inequality constraints, a modification of an existing algorithm for equality constrained problems can be used iteratively to solve the subproblems. Two test problems and two application problems are presented. The application examples include a rest-to-rest maneuver of a flexible structure and a constrained brachistochrone problem.
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    On the Convergence of the P-Algorithm for One-Dimensional Global Optimization of Smooth Functions (1999)
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    Journal of optimization theory and applications 102 (1999), S. 479-495 
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    Keywords: Global optimization ; Gaussian processes
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    Topics: Mathematics
    Notes: Abstract The Wiener process is a widely used statistical model for stochastic global optimization. One of the first optimization algorithms based on a statistical model, the so-called P-algorithm, was based on the Wiener process. Despite many advantages, this process does not give a realistic model for many optimization problems, particularly from the point of view of local behavior. In the present paper, a version of the P-algorithm is constructed based on a stochastic process with smooth sampling functions. It is shown that, in such a case, the algorithm has a better convergence rate than in the case of the Wiener process. A similar convergence rate is proved for a combination of the Wiener model-based P-algorithm with quadratic fit-based local search.
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    A Finite Algorithm for Global Minimization of Separable Concave Programs (1998)
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    Journal of global optimization 12 (1998), S. 1-36 
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    ISSN: 1573-2916
    Keywords: Global optimization ; concave programming ; branch-and-bound ; domainreduction
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    Topics: Mathematics
    Notes: Abstract Researchers first examined the problem of separable concave programming more than thirty years ago, making it one of the earliest branches of nonlinear programming to be explored. This paper proposes a new algorithm that finds the exact global minimum of this problem in a finite number of iterations. In addition to proving that our algorithm terminates finitely, the paper extends a guarantee of finiteness to all branch-and-bound algorithms for concave programming that (1) partition exhaustively using rectangular subdivisions and (2) branch on the incumbent solution when possible. The algorithm uses domain reduction techniques to accelerate convergence; it solves problems with as many as 100 nonlinear variables, 400 linear variables and 50 constraints in about five minutes on an IBM RS/6000 Power PC. An industrial application with 152 nonlinear variables, 593 linear variables, and 417 constraints is also solved in about ten minutes.
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    Optimal Control Problems via Exact Penalty Functions (1998)
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    Journal of global optimization 12 (1998), S. 215-223 
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    Keywords: Optimal control ; Exact penalization ; Directional differentiability ; Subdifferentiability ; Necessary optimality conditions ; Nonsmooth analysis
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    Topics: Mathematics
    Notes: Abstract The nonsmoothness is viewed by many people as at least an undesirable (if not unavoidable) property. Our aim here is to show that recent developments in Nonsmooth Analysis (especially in Exact Penalization Theory) allow one to treat successfully even some quite ‘smooth’ problems by tools of Nonsmooth Analysis and Nondifferentiable Optimization. Our approach is illustrated by one Classical Control Problem of finding optimal parameters in a system described by ordinary differential equations.
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    A Finite Algorithm for Globally Optimizing a Class of Rank-Two Reverse Convex Programs (1998)
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    Journal of global optimization 12 (1998), S. 247-265 
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    Keywords: Global optimization ; Reverse convex program ; Rank-two quasiconcave function ; Parametric simplex algorithm
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    Notes: Abstract In this paper, we propose an algorithm for solving a linear program with an additional rank-two reverse convex constraint. Unlike the existing methods which generate approximately optimal solutions, the algorithm provides a rigorous optimal solution to this nonconvex problem by a finite number of dual pivot operations. Computational results indicate that the algorithm is practical and can solve fairly large scale problems.
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    Testing the ℜ- strategy for a Reverse Convex Problem (1998)
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    Journal of global optimization 13 (1998), S. 61-74 
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    ISSN: 1573-2916
    Keywords: Global optimization ; Global optimality conditions ; Global search algorithm
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    Topics: Mathematics
    Notes: Abstract This paper is devoted to solving a reverse-convex problem. The approach presented here is based on Global Optimality Conditions. We propose a general conception of a Global Search Algorithm and develop each part of it. The results of numerical experiments with the dimension up to 400 are also given.
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    An Outer Approximation Algorithm for Generating All Efficient Extreme Points in the Outcome Set of a Multiple Objective Linear Programming Problem (1998)
    Springer
    Journal of global optimization 13 (1998), S. 1-24 
    Details
    ISSN: 1573-2916
    Keywords: Efficient set ; Global optimization ; Multiple objective linear programming ; Outer approximation ; Vector maximization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Various difficulties have been encountered in using decision set-based vector maximization methods to solve a multiple objective linear programming problem (MOLP). Motivated by these difficulties, some researchers in recent years have suggested that outcome set-based approaches should instead be developed and used to solve problem (MOLP). In this article, we present a finite algorithm, called the Outer Approximation Algorithm, for generating the set of all efficient extreme points in the outcome set of problem (MOLP). To our knowledge, the Outer Approximation Algorithm is the first algorithm capable of generating this set. As a by-product, the algorithm also generates the weakly efficient outcome set of problem (MOLP). Because it works in the outcome set rather than in the decision set of problem (MOLP), the Outer Approximation Algorithm has several advantages over decision set-based algorithms. It is also relatively easy to implement. Preliminary computational results for a set of randomly-generated problems are reported. These results tangibly demonstrate the usefulness of using the outcome set approach of the Outer Approximation Algorithm instead of a decision set-based approach.
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    URL: http://dx.doi.org/10.1023/A:1008215702611
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    On Global Optimality Conditions for Nonlinear Optimal Control Problems (1998)
    Springer
    Journal of global optimization 13 (1998), S. 109-122 
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    ISSN: 1573-2916
    Keywords: Optimal control ; Pontryagin maximum principle ; Global optimality
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let a trajectory and control pair $$(\bar x{\text{, }}\bar u{\text{)}}$$ maximize globally the functional g(x(T)) in the basic optimal control problem. Then (evidently) any pair (x,u) from the level set of the functional g corresponding to the value g( $$\bar x$$ (T)) is also globally optimal and satisfies the Pontryagin maximum principle. It is shown that this necessary condition for global optimality of $$(\bar x{\text{, }}\bar u{\text{)}}$$ turns out to be a sufficient one under the additional assumption of nondegeneracy of the maximum principle for every pair (x,u) from the above-mentioned level set. In particular, if the pair $$(\bar x{\text{, }}\bar u{\text{)}}$$ satisfies the Pontryagin maximum principle which is nondegenerate in the sense that for the Hamiltonian H, we have along the pair $$(\bar x{\text{, }}\bar u{\text{)}}$$ $$\mathop {{\text{max}}}\limits_u {\text{ }}H$$ ≢ $$\mathop {{\text{min}}}\limits_u {\text{ }}H$$ on [0,T], and if there is no another pair (x,u) such that g(x(T))=g( $$\bar x$$ (T)), then $$(\bar x{\text{, }}\bar u{\text{)}}$$ is a global maximizer.
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    URL: http://dx.doi.org/10.1023/A:1008211429313
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    A Crystal Growth Approach for Topographical Global Optimization (1998)
    Springer
    Journal of global optimization 13 (1998), S. 255-267 
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    ISSN: 1573-2916
    Keywords: Global optimization ; Topography graph ; Crystal growth
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new approach for topographical global minimization of a function f(x), x ∈ A ⊂ Rn by using sampled points in A is presented. The globally sampled points are firstly obtained by uniform random sampling or uniform sampling with threshold distances. The point with the lowest function value is used as the nucleus atom to start a crystal growth process. A first triangular nucleus includes the nucleus atom and two nearest points. Sequential crystal growth is continued for which a point next closest to the nucleus atom is bonded to the crystal by attaching to two nearest solidified points. A solidified point will be marked during the crystal growth process if any of two connected points has a lower function value. Upon completion of entire crystal growth process, all unmarked points are then used as starting points for subsequent local minimizations. Extension of the topographical algorithms to constrained problems is exercised by using penalty functions. Formulas for estimation on the number of sampled points for problems with an assumed number of local minima are provided. Results on three global minimization problems by two topographical algorithms are discussed.
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    URL: http://dx.doi.org/10.1023/A:1008288110360
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    Cutting Plane/Tabu Search Algorithms for Low Rank Concave Quadratic Programming Problems (1998)
    Springer
    Journal of global optimization 13 (1998), S. 225-240 
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    ISSN: 1573-2916
    Keywords: Low rank concave quadratic programming problem ; Tuy's cutting plane ; Rosen's cutting plane ; Global optimization ; Tabu-search ; Heuristic algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we will propose an efficient heuristic algorithm for solving concave quadratic programming problems whose rank of the objective function is relatively small. This algorithm is a combination of Tuy's cutting plane to eliminate the feasible region and a kind of tabu-search method to find a ‘good’ vertex. We first generate a set of V of vertices and select one of these vertices as a starting point at each step, and apply tabu-search and Tuy's cutting plane algorithm where the list of tabu consists of those vertices eliminated by cutting planes and those newly generated vertices by cutting planes. When all vertices of the set V are eliminated, the algorithm is terminated. This algorithm need not converge to a global minimum, but it can work very well when the rank is relatively small (up to seven). The incumbent solutions are in fact globally optimal for all tested problems. We also propose an alternative algorithm by incorporating Rosen's hyperrectangle cut. This algorithm is more efficient than the combination of Tuy's cutting plane and tabu-search.
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    URL: http://dx.doi.org/10.1023/A:1008230825152
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    An algorithm for solving global optimization problems with nonlinear constraints (1995)
    Springer
    Journal of global optimization 7 (1995), S. 407-419 
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    ISSN: 1573-2916
    Keywords: Global optimization ; nonlinear constraints ; local tuning ; index scheme ; global convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we propose an algorithm using only the values of the objective function and constraints for solving one-dimensional global optimization problems where both the objective function and constraints are Lipschitzean and nonlinear. The constrained problem is reduced to an unconstrained one by the index scheme. To solve the reduced problem a new method with local tuning on the behavior of the objective function and constraints over different sectors of the search region is proposed. Sufficient conditions of global convergence are established. We also present results of some numerical experiments.
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    URL: http://dx.doi.org/10.1007/BF01099650
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    An algorithm for maximizing a convex function over a simple set (1996)
    Springer
    Journal of global optimization 8 (1996), S. 379-391 
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    ISSN: 1573-2916
    Keywords: Global optimization ; optimality condition ; convex function ; numerical algorithm ; simple set
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The problem of maximizing a convex function on a so-called simple set is considered. Based on the optimality conditions [19], an algorithm for solving the problem is proposed. This numerical algorithm is shown to be convergent. The proposed algorithm has been implemented and tested on a variety of test problems.
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    URL: http://dx.doi.org/10.1007/BF02403999
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    Optimal random non-adaptive algorithm for global optimization of Brownian motion (1996)
    Springer
    Journal of global optimization 8 (1996), S. 81-90 
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    ISSN: 1573-2916
    Keywords: Global optimization ; average performance ; Brownian motion
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we study random non-adaptive algorithms for finding the maximum of a continuous function on the unit interval. We compare the average performance of different algorithms under the assumption of Wiener measure on the space of continuous functions. Placing the observations independently according to a Beta(2/3,2/3) density function is shown to be the optimal random non-adaptive algorithm. The performance is compared with other random and deterministic non-adaptive algorithms.
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    URL: http://dx.doi.org/10.1007/BF00229303
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    Finding all solutions of nonlinearly constrained systems of equations (1995)
    Springer
    Journal of global optimization 7 (1995), S. 143-182 
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    ISSN: 1573-2916
    Keywords: Global optimization ; nonlinear systems of equations ; all solutions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new approach is proposed for finding allε-feasible solutions for certain classes of nonlinearly constrained systems of equations. By introducing slack variables, the initial problem is transformed into a global optimization problem (P) whose multiple global minimum solutions with a zero objective value (if any) correspond to all solutions of the initial constrained system of equalities. Allε-globally optimal points of (P) are then localized within a set of arbitrarily small disjoint rectangles. This is based on a branch and bound type global optimization algorithm which attains finiteε-convergence to each of the multiple global minima of (P) through the successive refinement of a convex relaxation of the feasible region and the subsequent solution of a series of nonlinear convex optimization problems. Based on the form of the participating functions, a number of techniques for constructing this convex relaxation are proposed. By taking advantage of the properties of products of univariate functions, customized convex lower bounding functions are introduced for a large number of expressions that are or can be transformed into products of univariate functions. Alternative convex relaxation procedures involve either the difference of two convex functions employed in αBB [23] or the exponential variable transformation based underestimators employed for generalized geometric programming problems [24]. The proposed approach is illustrated with several test problems. For some of these problems additional solutions are identified that existing methods failed to locate.
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    URL: http://dx.doi.org/10.1007/BF01097059
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    αBB: A global optimization method for general constrained nonconvex problems (1995)
    Springer
    Journal of global optimization 7 (1995), S. 337-363 
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    ISSN: 1573-2916
    Keywords: Global optimization ; constrained optimization ; convex relaxation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A branch and bound global optimization method,αBB, for general continuous optimization problems involving nonconvexities in the objective function and/or constraints is presented. The nonconvexities are categorized as being either of special structure or generic. A convex relaxation of the original nonconvex problem is obtained by (i) replacing all nonconvex terms of special structure (i.e. bilinear, fractional, signomial) with customized tight convex lower bounding functions and (ii) by utilizing the α parameter as defined in [17] to underestimate nonconvex terms of generic structure. The proposed branch and bound type algorithm attains finiteε-convergence to the global minimum through the successive subdivision of the original region and the subsequent solution of a series of nonlinear convex minimization problems. The global optimization method,αBB, is implemented in C and tested on a variety of example problems.
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    URL: http://dx.doi.org/10.1007/BF01099647
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    Global optimization with a limited solution time (1996)
    Springer
    Journal of global optimization 8 (1996), S. 413-427 
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    ISSN: 1573-2916
    Keywords: Global optimization ; random methods ; heuristic solution strategy ; local search ; limited solution time
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper the box constrained global optimization problem in presence of a limited solution time is considered. A method is studied based on a combination of multistart and singlestart which implies a decision sequence on the number of random points to be generated. Search strategies are numerically illustrated. Criteria are introduced to measure the performance of solution methods for the problem class. Moreover, the performance of search strategies, specifically the efficiency of generating random points is analyzed.
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    URL: http://dx.doi.org/10.1007/BF02404001
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    Aspiration Based Simulated Annealing Algorithm (1997)
    Springer
    Journal of global optimization 11 (1997), S. 181-191 
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    ISSN: 1573-2916
    Keywords: Global optimization ; continuous variables ; aspiration value ; simulated annealing ; stochastic
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An aspiration based simulated annealing algorithm for continuousvariables has been proposed. The new algorithm is similar to the one givenby Dekkers and Aarts (1991) except that a kind of memory is introduced intothe procedure with a self-regulatory mechanism. The algorithm has beenapplied to a set of standard global optimization problems and a number ofmore difficult, complex, practical problems and its performance comparedwith that of the algorithm of Dekkers and Aarts (1991). The new algorithmappears to offer a useful alternative to some of the currently availablestochastic algorithms for global optimization.
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    URL: http://dx.doi.org/10.1023/A:1008202703889
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    A Reduced Space Branch and Bound Algorithm for Global optimization (1997)
    Springer
    Journal of global optimization 11 (1997), S. 287-311 
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    ISSN: 1573-2916
    Keywords: Global optimization ; branch and bound ; reduced space ; convexenvelope
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A general class of branch and bound algorithms forsolving a wide class of nonlinear programs with branching only in asubset of the problem variables is presented. By reducing the dimension of thesearch space, this technique may dramatically reduce the number ofiterations and time required for convergence to ∈ tolerancewhile retaining proven exact convergence in the infinite limit. Thispresentation includes specifications of the class of nonlinearprograms, a statement of a class of branch and bound algorithms, aconvergence proof, and motivating examples with results.
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    URL: http://dx.doi.org/10.1023/A:1008212418949
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    A Numerical Comparison of Some Modified Controlled Random Search Algorithms (1997)
    Springer
    Journal of global optimization 11 (1997), S. 377-385 
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    ISSN: 1573-2916
    Keywords: Global optimization ; β-distribution ; controlled random search
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we propose a new version of the Controlled Random Search(CRS) algorithm of Price. The new algorithmhas been tested on thirteen global optimization test problems. Numericalexperiments indicate that the resulting algorithm performs considerablybetter than the earlier versions of the CRS algorithms. The algorithm,therefore, could offer a reasonable alternative to many currently availablestochastic algorithms, especially for problems requiring ’direct search‘type methods. Also a classification of the CRS algorithms is made based on’global technique‘ – ’local technique‘ and the relative performance ofclasses is numerically explored.
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    URL: http://dx.doi.org/10.1023/A:1008236920512
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    The Chebyshev Hyperplane Optimization Problem (1997)
    Springer
    Journal of global optimization 11 (1997), S. 361-376 
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    ISSN: 1573-2916
    Keywords: Global optimization ; curve fitting ; Chebyshev approximation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider the following problem. Given a finite set of pointsy j in $$\mathbb{R}^n $$ we want to determine a hyperplane H such that the maximum Euclidean distance betweenH and the pointsy j is minimized. This problem(CHOP) is a non-convex optimization problem with a special structure. Forexample, all local minima can be shown to be strongly unique. We present agenericity analysis of the problem. Two different global optimizationapproaches are considered for solving (CHOP). The first is a Lipschitzoptimization method; the other a cutting plane method for concaveoptimization. The local structure of the problem is elucidated by analysingthe relation between (CHOP) and certain associated linear optimizationproblems. We report on numerical experiments.
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    URL: http://dx.doi.org/10.1023/A:1008220431204
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    A Simplicial Branch-and-Bound Method for Solving Nonconvex All-Quadratic Programs (1998)
    Springer
    Journal of global optimization 13 (1998), S. 417-432 
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    ISSN: 1573-2916
    Keywords: Branch-and-bound ; Global optimization ; Indefinite quadratic optimization under indefinite quadratic constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we present an algorithm for solving nonconvex quadratically constrained quadratic programs (all-quadratic programs). The method is based on a simplicial branch-and-bound scheme involving mainly linear programming subproblems. Under the assumption that a feasible point of the all-quadratic program is known, the algorithm guarantees an ε-approximate optimal solution in a finite number of iterations. Computational experiments with an implementation of the procedure are reported on randomly generated test problems. The presented algorithm often outperforms a comparable rectangular branch-and-bound method.
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    URL: http://dx.doi.org/10.1023/A:1008377529330
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    Minimizing Increasing Star-shaped Functions Based on Abstract Convexity (1999)
    Springer
    Journal of global optimization 15 (1999), S. 19-39 
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    ISSN: 1573-2916
    Keywords: Abstract convexity ; Global optimization ; Increasing function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study a broad class of increasing non-convex functions whose level sets are star shaped with respect to infinity. We show that these functions (we call them ISSI functions) are abstract convex with respect to the set of min-type functions and exploit this fact for their minimization. An algorithm is proposed for solving global optimization problems with an ISSI objective function and its numerical performance is discussed.
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    URL: http://dx.doi.org/10.1023/A:1008344317743
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    Parallel Information Algorithm with Local Tuning for Solving Multidimensional GO Problems (1999)
    Springer
    Journal of global optimization 15 (1999), S. 157-167 
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    ISSN: 1573-2916
    Keywords: Global optimization ; Parallel computations ; Local tuning
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we propose a new parallel algorithm for solving global optimization (GO) multidimensional problems. The method unifies two powerful approaches for accelerating the search: parallel computations and local tuning on the behavior of the objective function. We establish convergence conditions for the algorithm and theoretically show that the usage of local information during the global search permits to accelerate solving the problem significantly. Results of numerical experiments executed with 100 test functions are also reported.
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    URL: http://dx.doi.org/10.1023/A:1008372702319
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    Time-Varying Linear-Quadratic Control (1999)
    Springer
    Journal of optimization theory and applications 100 (1999), S. 515-525 
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    ISSN: 1573-2878
    Keywords: Optimal control ; linear-quadratic control ; time-varying feedback
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper discusses the performance of controlled linear dynamic systems that use time-varying feedforward signals and time-varying linear-quadratic (LQ) feedback gains. Such a time-varying LQ controller can bring a dynamic system to a desired final state in roughly half the time required by a time-invariant LQ controller, since it pushes at both ends, i.e., it uses significant control effort near the end of the maneuver, as well as at the beginning, to meet the specified end conditions; there is no overshoot and no settling time. This requires a more complex controller and some care with the high gains near the final time. A MATLAB3 code is listed that synthesizes and simulates zero-order-hold time-varying LQ controllers. The precision landing of a helicopter using four controls is treated as an example.
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    URL: http://dx.doi.org/10.1023/A:1022682305644
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    Neighborhood Turnpike Theorem for Continuous-Time Optimization Models (1999)
    Springer
    Journal of optimization theory and applications 101 (1999), S. 651-676 
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    ISSN: 1573-2878
    Keywords: Optimal control ; dynamic models ; optimal growth ; turnpike theorems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A neighborhood turnpike theorem is proved for continuous-time, infinite-horizon optimization models with positive discounting. Our approach is a primal one and no differentiability assumption is made. The basic hypothesis is a condition of uniform concavity at the point defining the undiscounted steady state. The main novelty here is that we formulate the theorem by taking the undiscounted steady state as the turnpike.
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    URL: http://dx.doi.org/10.1023/A:1021794221688
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    Gradient-flow approach for computing a nonlinear-quadratic optimal-output feedback gain matrix (1995)
    Springer
    Journal of optimization theory and applications 85 (1995), S. 75-96 
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    ISSN: 1573-2878
    Keywords: Optimal control ; infinite horizon ; nonlinear-quadratic regulator problem ; linear state feedback ; approximate problems ; gradient-flow based algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, an approach is proposed for solving a nonlinear-quadratic optimal regulator problem with linear static state feedback and infinite planning horizon. For such a problem, approximate problems are introduced and considered, which are obtained by combining a finite-horizon problem with an infinite-horizon linear problem in a certain way. A gradient-flow based algorithm is derived for these approximate problems. It is shown that an optimal solution to the original problem can be found as the limit of a sequence of solutions to the approximate problems. Several important properties are obtained. For illustration, two numerical examples are presented.
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    URL: http://dx.doi.org/10.1007/BF02192300
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    Shooting method for the numerical solution of optimal control problems with bounded state variables (1996)
    Springer
    Journal of optimization theory and applications 89 (1996), S. 351-372 
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    ISSN: 1573-2878
    Keywords: Optimal control ; bounded state ; inequality constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abtract Various methods have been proposed for the numerical solution of optimal control problems with bounded state variables. In this paper, a new method is put forward and compared with two other methods, one of which makes use of adjoint variables whereas the other does not. Some conclusions are drawn on the usefulness of the three methods involved.
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    URL: http://dx.doi.org/10.1007/BF02192534
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    Pathways to hopf bifurcations in dynamic continuous-time optimization problems (1996)
    Springer
    Journal of optimization theory and applications 91 (1996), S. 299-320 
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    ISSN: 1573-2878
    Keywords: Optimal control ; limit cycles ; Hopf bifurcation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The purpose of this paper is to characterize pathways to Hopf bifurcation in continuous time, concave, two-dimensional optimal control models. It is shown that essentially two pathways exist: control-state interaction and growth. The knowledge of such pathways provides a criterion at the stage of modelling on the potential complexity of optimal trajectories.
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    An Optimal Control Problem for Flows with Discontinuities (1997)
    Springer
    Journal of optimization theory and applications 94 (1997), S. 273-309 
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    ISSN: 1573-2878
    Keywords: Optimal control ; Euler flow equations ; sequential quadratic programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we study a design problem for a duct flow with a shock. The presence of the shock causes numerical difficulties. Good shock-capturing schemes with low continuity properties often cannot be combined successfully with efficient optimization methods requiring smooth functions. A remedy studied in this paper is to introduce the shock location as an explicit variable. This allows one to fit the shock and yields a problem with sufficiently smooth functions. We prove the existence of optimal solutions, Fréchet differentiability, and the existence of Lagrange multipliers. In the second part, we introduce and investigate the discrete problem and study the relations between the optimality conditions for the infinite-dimensional problem and the discretized one. This reveals important information for the numerical solution of the problem. Numerical examples are given to demonstrate the theoretical findings.
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    URL: http://dx.doi.org/10.1023/A:1022616327742
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    Properties of the Bellman Function in Time-Optimal Control Problems (1997)
    Springer
    Journal of optimization theory and applications 94 (1997), S. 155-168 
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    ISSN: 1573-2878
    Keywords: Optimal control ; Bellman function ; value function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, time-optimal control problems with closed terminal sets are considered. We give conditions which guarantee the Bellman function to be Hölder and Lipschitz continuous. We then prove that the condition for Lipschitz continuity is also necessary.
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    URL: http://dx.doi.org/10.1023/A:1022615921015
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    Application of Stochastic Global Optimization Algorithms to Practical Problems (1997)
    Springer
    Journal of optimization theory and applications 95 (1997), S. 545-563 
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    ISSN: 1573-2878
    Keywords: Global optimization ; real life problems ; pig liver likelihood function ; many-body potential function ; tank reactor ; optimal control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We describe global optimization problems from three different fields representing many-body potentials in physical chemistry, optimal control of a chemical reactor, and fitting a statistical model to empirical data. Historical background for each of the problems as well as the practical significance of the first two are given. The problems are solved by using eight recently developed stochastic global optimization algorithms representing controlled random search (4 algorithms), simulated annealing (2 algorithms), and clustering (2 algorithms). The results are discussed, and the importance of global optimization in each respective field is focused.
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    URL: http://dx.doi.org/10.1023/A:1022617804737
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