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  • Articles  (43)
  • Articles: DFG German National Licenses  (43)
  • Optimal control  (28)
  • nonlinear programming  (15)
  • 2010-2014
  • 1980-1984  (43)
  • 1950-1954
  • Mathematics  (43)
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  • Articles  (43)
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  • Articles: DFG German National Licenses  (43)
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  • 1
    Electronic Resource
    Electronic Resource
    A globally convergent, implementable multiplier method with automatic penalty limitation (1980)
    Springer
    Applied mathematics & optimization 6 (1980), S. 335-360 
    Details
    ISSN: 1432-0606
    Keywords: nonlinear programming ; multiplier methods ; penalty methods ; global convergence ; penalty limitation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper deals with penalty function and multiplier methods for the solution of constrained nonconvex nonlinear programming problems. Starting from an idea introduced several years ago by Polak, we develop a class of implementable methods which, under suitable assumptions, produce a sequence of points converging to a strong local minimum for the problem, regardless of the location of the initial guess. In addition, for sequential minimization type multiplier methods, we make use of a rate of convergence result due to Bertsekas and Polyak, to develop a test for limiting the growth of the penalty parameter and thereby prevent ill-conditioning in the resulting sequence of unconstrained optimization problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01442901
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  • 2
    Electronic Resource
    Electronic Resource
    Analysis of a sterile insect release model with predation (1982)
    Springer
    Journal of mathematical biology 16 (1982), S. 33-48 
    Details
    ISSN: 1432-1416
    Keywords: Sterile insect release ; Predation ; Stability ; Limit cycles ; Optimal control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A model for the sterile insect release method of pest control in which the target species is under predatory or parasitic regulation is analyzed. The equations are nondimensionalized and the rescaled parameters are interpreted. There are four types of equilibria, whose existence and stability depend on which of ten regions of parameter space contain the rescaled parameters, and in turn give minimal release rates to achieve eradication of the pest. In at least one region, Hopf bifurcation theory shows the existence of limit cycles, but they are found to be unstable. In addition, the optimal release rate to minimize a total cost functional for pest control by the sterile release method is studied. Both approaches show that when predation accounts for a large fraction of the natural deaths, the necessary release rate and total cost are higher than for weak predation. If the predators are removed without being replaced by any other source of mortality, the cost rises in all cases but rises much more dramatically for cases with strong predation. A definite danger of the sterile release method when some predatory control exists is that the predators are frequently driven extinct before the prey, so that the target species could explode to much higher levels and be more difficult to eradicate again after the sterile release is terminated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275159
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  • 3
    Electronic Resource
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    Some remarks on the existence of optimal controls for quasilinear systems (1981)
    Springer
    Journal of optimization theory and applications 33 (1981), S. 393-399 
    Details
    ISSN: 1573-2878
    Keywords: Optimal control ; nonlinear systems ; existence theorems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let a quasilinear control system having the state space $$\bar X \subseteq R^n $$ be governed by the vector differential equation $$\dot x = G(u(t))x,$$ wherex(0) =x 0 andU is the family of all bounded measurable functions from [0,T] intoU, a compact and convex subset ofR m.LetG:U ⇑R be a bounded measurable nonlinear function, such thatG(U) is compact and convex.G −1 can be convex onG(U) or concave. The main results of the paper establish the existence of a controlu ∈U which minimizes the cost functional $$I(u) = \int_0^T {L(u(t))x(t)dt,} $$ whereL(·) is convex. A practical example of application for chemical reactions is worked out in detail.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00935251
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  • 4
    Electronic Resource
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    A limiting Lagrangian for infinitely constrained convex optimization inR n (1981)
    Springer
    Journal of optimization theory and applications 33 (1981), S. 479-495 
    Details
    ISSN: 1573-2878
    Keywords: Lagrangians ; nonlinear programming ; Kuhn-Tucker theory ; convex optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For convex optimization inR n,we show how a minor modification of the usual Lagrangian function (unlike that of the augmented Lagrangians), plus a limiting operation, allows one to close duality gaps even in the absence of a Kuhn-Tucker vector [see the introductory discussion, and see the discussion in Section 4 regarding Eq. (2)]. The cardinality of the convex constraining functions can be arbitrary (finite, countable, or uncountable). In fact, our main result (Theorem 4.3) reveals much finer detail concerning our limiting Lagrangian. There are affine minorants (for any value 0〈θ≤1 of the limiting parameter θ) of the given convex functions, plus an affine form nonpositive onK, for which a general linear inequality holds onR nAfter substantial weakening, this inequality leads to the conclusions of the previous paragraph. This work is motivated by, and is a direct outgrowth of, research carried out jointly with R. J. Duffin.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00935754
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  • 5
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    A new augmented Lagrangian function for inequality constraints in nonlinear programming problems (1982)
    Springer
    Journal of optimization theory and applications 36 (1982), S. 495-519 
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    ISSN: 1573-2878
    Keywords: Optimization ; nonlinear programming ; Numerical methods ; computational methods ; augmented Lagrangian functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, a new augmented Lagrangian function is introduced for solving nonlinear programming problems with inequality constraints. The relevant feature of the proposed approach is that, under suitable assumptions, it enables one to obtain the solution of the constrained problem by a single unconstrained minimization of a continuously differentiable function, so that standard unconstrained minimization techniques can be employed. Numerical examples are reported.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00940544
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  • 6
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    Sufficient conditions for the optimal control of a class systems with continuous lags (1982)
    Springer
    Journal of optimization theory and applications 38 (1982), S. 153-154 
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    ISSN: 1573-2878
    Keywords: Optimal control ; maximum principle ; sufficient conditions ; integro-differential equations ; path constraints ; economic applications
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Notational errors in Theorem 5.2 of Ref. 1 are corrected.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934332
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  • 7
    Electronic Resource
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    On the optimal control problem of a single-server queueing system (1983)
    Springer
    Journal of optimization theory and applications 41 (1983), S. 317-325 
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    ISSN: 1573-2878
    Keywords: Optimal control ; queueing theory ; phase type distributions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For a single-server queueing system (with a finite waiting room) with phase type arrivals and exponential service times, an optimal control for the service rate is derived. This generalizes the result of Scott and Jefferson for theM/M/1/1 queueing model.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00935227
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  • 8
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    Near-optimum design of nonstationary linear systems with state and control delays (1980)
    Springer
    Journal of optimization theory and applications 30 (1980), S. 73-88 
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    ISSN: 1573-2878
    Keywords: Optimal control ; suboptimal control ; time-delay systems ; linear systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A computationally efficient method based on a sensitivity approach is developed to obtain suboptimal control for nonstationary linear systems with multiple state and control delays and with quadratic cost. The proposed suboptimal control includes some truncated series whose terms are calculated in a recursive manner from nondelay system optimization.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934590
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  • 9
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    Linearized ridge-path method for function minimization (1980)
    Springer
    Journal of optimization theory and applications 30 (1980), S. 161-179 
    Details
    ISSN: 1573-2878
    Keywords: Optimization techniques ; nonlinear programming ; direct methods ; numerical methods ; conjugate directions ; nongradient methods ; ridge-path methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A modification based on a linearization of a ridge-path optimization method is presented. The linearized ridge-path method is a nongradient, conjugate direction method which converges quadratically in half the number of search directions required for Powell's method of conjugate directions. The ridge-path method and its modification are compared with some basic algorithms, namely, univariate method, steepest descent method, Powell's conjugate direction method, conjugate gradient method, and variable-metric method. The assessment indicates that the ridge-path method, with modifications, could present a promising technique for optimization.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934494
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  • 10
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    Suboptimal control of systems with multiple delays (1980)
    Springer
    Journal of optimization theory and applications 30 (1980), S. 621-633 
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    ISSN: 1573-2878
    Keywords: Optimal control ; time-delay systems ; linear systems ; multiple-delay systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The application of Pontryagin's maximum principle to the optimization of linear systems with time delays results in a system of coupled two-point boundary-value problems involving both delay and advance terms. The exact solution of this system of TPBV problems is extremely difficult, if not impossible. In this paper, a fast-converging iterative approach is developed for obtaining the suboptimal control for nonstationary linear systems with multiple state and control delays and with quadratic cost. At each step of the proposed method, a linear nondelay system with an extra perturbing input must be optimized. The procedure can be extended for the optimization of nonlinear systems with multiple time-varying delays, provided that some of the nonlinearities satisfy the Lipschitz condition.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01686725
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  • 11
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    Solution of nonlinear optimal control problems in Hilbert spaces by means of linear programming techniques (1980)
    Springer
    Journal of optimization theory and applications 30 (1980), S. 643-661 
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    ISSN: 1573-2878
    Keywords: Optimal control ; measures ; Hilbert spaces ; linear programming ; approximation techniques
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Optimal control problems in Hilbert spaces are considered in a measure-theoretical framework. Instead of minimizing a functional defined on a class of admissible trajectory-control pairs, we minimize one defined on a set of measures; this set is defined by the boundary conditions and the differential equation of the problem. The new problem is an infinite-dimensionallinear programming problem; it is shown that it is possible to approximate its solution by that of a finite-dimensional linear program of sufficiently high dimensions, while this solution itself can be approximated by a trajectory-control pair. This pair may not be strictly admissible; if the dimensionality of the finite-dimensional linear program and the accuracy of the computations are high enough, the conditions of admissibility can be said to be satisfied up to any given accuracy. The value given by this pair to the functional measuring the performance criterion can be about equal to theglobal infimum associated with the classical problem, or it may be less than this number. It appears that this method may become a useful technique for the computation of optimal controls, provided the approximations involved are acceptable.
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    URL: http://dx.doi.org/10.1007/BF01686727
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  • 12
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    Computational experience with Davidon's least-square algorithm (1980)
    Springer
    Journal of optimization theory and applications 31 (1980), S. 27-39 
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    ISSN: 1573-2878
    Keywords: Least-square methods ; variable-metric methods ; Levenberg-Marquardt methods ; nonlinear programming ; testing algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Computational results are presented for Davidon's new least-square algorithm. Computational experience with this algorithm is reported which motivated the development of a production code version of the algorithm. Several heuristic modifications, which have been added, are described. Fifteen zero-residual test problems have been used in comparing the new production code version with two established versions of the Levenberg-Marquardt algorithm. The production code version of Davidon's least-square algorithm performed faster and used less function evaluations than the Levenberg-Marquardt algorithm in almost every case of the test problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934786
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  • 13
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    Junction phenomena for optimal control with state-variable inequality constraints of third order (1980)
    Springer
    Journal of optimization theory and applications 31 (1980), S. 85-99 
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    ISSN: 1573-2878
    Keywords: Optimal control ; junction conditions ; calculus of variations ; inequality constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is known that extremal arcs governed by inequality constraints of third order (constraint relations that must be differentiated three times to generate a control equation) cannot join an unconstrained arc, except in special cases. But a control problem is exhibited, for which every extremal includes a constrained arc of third order. The constrained arc joins the end of an infinite sequence of consecutive unconstrained arcs of finite total duration. Evidence (but not proof) is given that this phenomenon is typical, rather than exceptional. An analogous phenomenon is well known for optimal control problems with singular arcs of second order.
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    URL: http://dx.doi.org/10.1007/BF00934790
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  • 14
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    Constrained nonzero-sum games with partially controllable strategies (1980)
    Springer
    Journal of optimization theory and applications 31 (1980), S. 361-371 
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    ISSN: 1573-2878
    Keywords: Nash-equilibrium solutions ; partially controllable strategies ; nonlinear programming ; complementary eigenvalue problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The present paper deals with a class of nonzero-sum, two-person games with finite strategies when there are constraints on the strategies selected by the players. The constraints arise due to the subjective difficulty that each player often has in assigning to the states probabilities with which he is completely satisfied, and the model specifies how much each player must perturb his initial probability estimate in order to change his maximum utility alternative from the alternative originally best under the initial estimate. It is shown that the Nash-equilibrium solution of this class of nonzero-sum games can be characterized by an equivalent nonlinear program which leads in some cases to a pair of complementary eigenvalue problems. Applications to normal or approximate solutions of linear programming problems are also indicated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01262978
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  • 15
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    Existence and global asymptotic stability of optimal trajectories for a class of infinite-horizon, nonconvex systems (1980)
    Springer
    Journal of optimization theory and applications 31 (1980), S. 515-533 
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    ISSN: 1573-2878
    Keywords: Optimal control ; economic systems ; stability ; existence theory ; turnpike property
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Sufficient conditions for the existence of optimal trajectories and for the global asymptotic stability of these trajectories are given for a class of nonconvex and nonautonomous systems controlled over an infinite-time horizon. The concept ofG-supported trajectory is introduced. It is shown that, under some assumptions, aG-supported trajectory is overtaking and is globally asymptotically stable. The concept of overtaking trajectory has been previously defined as a notion of optimality on an infinite-time domain. For autonomous systems, under weaker conditions, one guarantees the existence of weakly overtaking trajectories. Finally, it is shown howG-supported trajectories can be obtained, and an application to the study of a pre-predator ecosystem optimally harvested is sketched.
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    URL: http://dx.doi.org/10.1007/BF00934475
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  • 16
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    On optimal control problems with state constraints (1980)
    Springer
    Journal of optimization theory and applications 32 (1980), S. 75-80 
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    ISSN: 1573-2878
    Keywords: Optimal control ; state constraints ; multipliers ; perturbation theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is concerned with necessary conditions for a general optimal control problem developed by Russak and Tan. It is shown that, in most cases, a further relation between the multipliers holds. This result is of interest in particular for the investigation of perturbations of the state constraint.
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    URL: http://dx.doi.org/10.1007/BF00934843
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  • 17
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    A property of an autonomous minimax optimal control problem (1980)
    Springer
    Journal of optimization theory and applications 32 (1980), S. 81-87 
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    ISSN: 1573-2878
    Keywords: Optimal control ; minimax problems ; maximum principle ; Hamiltonian function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A control system $$\dot x = f\left( {x,u} \right)$$ ,u) with cost functional $$\mathop {ess \sup }\limits_{T0 \leqslant t \leqslant T1} G\left( {x\left( t \right),u\left( t \right)} \right)$$ is considered. For an optimal pair $$\left( {\bar x\left( \cdot \right),\bar u\left( \cdot \right)} \right)$$ ,ū(·)), there is a maximum principle of the form $$\eta \left( t \right)f\left( {\bar x\left( t \right),\bar u\left( t \right)} \right) = \mathop {\max }\limits_{u \in \Omega \left( t \right)} \eta \left( t \right)f\left( {\bar x\left( t \right),u} \right).$$ By means of this fact, it is shown that $$\eta \left( t \right)f\left( {\bar x\left( t \right),\bar u\left( t \right)} \right)$$ is equal to a constant almost everywhere.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934844
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  • 18
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    An exact penalty function algorithm for optimal control problems with control and terminal equality constraints, part 1 (1980)
    Springer
    Journal of optimization theory and applications 32 (1980), S. 211-246 
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    ISSN: 1573-2878
    Keywords: Optimal control ; control constraints ; terminal equality constraints ; exact penalty function ; convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The presence of control constraints, because they are nondifferentiable in the space of control functions, makes it difficult to cope with terminal equality constraints in optimal control problems. Gradient-projection algorithms, for example, cannot be employed easily. These difficulties are overcome in this paper by employing an exact penalty function to handle the cost and terminal equality constraints and using the control constraints to define the space of permissible search directions in the search-direction subalgorithm. The search-direction subalgorithm is, therefore, more complex than the usual linear program employed in feasible-directions algorithms. The subalgorithm approximately solves a convex optimal control problem to determine the search direction; in the implementable version of the algorithm, the accuracy of the approximation is automatically increased to ensure convergence.
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    URL: http://dx.doi.org/10.1007/BF00934725
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  • 19
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    Sufficient conditions for asymptotic optimal control (1980)
    Springer
    Journal of optimization theory and applications 32 (1980), S. 307-325 
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    ISSN: 1573-2878
    Keywords: Optimal control ; asymptotic optimal control ; calculus of variations ; Lagrange problems ; global minima ; absolute minima ; necessary conditions ; sufficient conditions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For a selected family of Lagrange-type control problems involving a nonnegative integral costJ T (y,u) over the interval [0,T], 0〈T〈∞, with system conditions consisting of differential inequalities and/or equalities, the following material is treated: (i) a resumé of relevant necessary conditions and sufficient conditions for a pair (y T ,u T ) to minimizeJ T (y,u); (ii) conditions sufficient for the convergence asT→∞ of minimizing pairs (y T ,u T ) over [0,T] to a limit pair (y ∞,u ∞) over the infinite-time interval [0, ∞); (iii) conditions sufficient for (y ∞,u ∞) to minimize the costJ ∞(y,u) over [0, ∞); and (iv) conditions sufficient for the optimal cost per unit timeJ T (y T ,u T )/T to have a limit asT→∞.
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    URL: http://dx.doi.org/10.1007/BF00934555
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  • 20
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    An exact penalty function algorithm for optimal control problems with control and terminal equality constraints, part 2 (1980)
    Springer
    Journal of optimization theory and applications 32 (1980), S. 345-364 
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    ISSN: 1573-2878
    Keywords: Optimal control ; control constraints ; terminal equality constraints ; exact penalty functions ; convergence ; relaxed control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In Part 1 of this paper, implementable and conceptual versions of an algorithm for optimal control problems with control constraints and terminal equality constraints were presented. It was shown that anyL ∞ accumulation points of control sequences generated by the algorithms satisfy necessary conditions of optimality. Since such accumulation points need not exist, it is shown in this paper that control sequences generated by the algorithms always have accumulation points in the sense of control measure, and these accumulation points satisfy optimality conditions for the corresponding relaxed control problem.
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    URL: http://dx.doi.org/10.1007/BF00934557
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  • 21
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    Generalized arcwise-connected functions and characterizations of local-global minimum properties (1980)
    Springer
    Journal of optimization theory and applications 32 (1980), S. 407-425 
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    ISSN: 1573-2878
    Keywords: Generalized convexity ; global minimality ; nonlinear programming ; nonconvex programming ; optimization theorems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, new classes of generalized convex functions are introduced, extending the concepts of quasi-convexity, pseudoconvexity, and their associate subclasses. Functions belonging to these classes satisfy certain local-global minimum properties. Conversely, it is shown that, under some mild regularity conditions, functions for which the local-global minimum properties hold must belong to one of the classes of functions introduced.
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    URL: http://dx.doi.org/10.1007/BF00934030
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  • 22
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    On the order of singular optimal control problems (1980)
    Springer
    Journal of optimization theory and applications 32 (1980), S. 479-489 
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    ISSN: 1573-2878
    Keywords: Optimal control ; singular control ; junction conditions ; singular control order
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In singular optimal control problems, the functional form of the optimal control function is usually determined by solving the algebraic equation which results by successively differentiating the switching function until the control appears explicitly. This process defines the order of the singular problem. Order-related results are developed for singular linear-quadratic problems and for a bilinear example which gives new insights into the relationship between singular problem order and singular are order.
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    URL: http://dx.doi.org/10.1007/BF00934035
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  • 23
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    On the general inverse problem of optimal control theory (1980)
    Springer
    Journal of optimization theory and applications 32 (1980), S. 491-497 
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    ISSN: 1573-2878
    Keywords: Optimal control ; inverse problems ; dynamic programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The general inverse problem of optimal control is considered from a dynamic programming point of view. Necessary and sufficient conditions are developed which two integral criteria must satisfy if they are to yield the same optimal feedback law, the dynamics being fixed. Specializing to the linear-quadratic case, it is shown how the general results given here recapture previously obtained results for quadratic criteria with linear dynamics.
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    URL: http://dx.doi.org/10.1007/BF00934036
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  • 24
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    Realistic feedback control of two interconnected turbogenerators (1984)
    Springer
    Journal of optimization theory and applications 42 (1984), S. 103-141 
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    ISSN: 1573-2878
    Keywords: Optimal control ; realistic feedback control ; interconnected turbogenerators ; torque and voltage control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, the optimal control of a system with two identical interconnected turbogenerators, which are connected to an infinite bus, is considered. The alternators are controlled through a linear feedback of the state variables. The feedback parameters are obtained by solving a nonlinear, two-point boundary-value problem. The values obtained for these parameters depend on the strength and duration of the disturbance, since the model is nonlinear, contrary to the usual feedback control of a linear model. In contrast to the model used in Ref. 1, the model used here includes the transfer function of the governors, the turbines, and the voltage regulators.
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    URL: http://dx.doi.org/10.1007/BF00934135
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  • 25
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    Convergence of a strong variational algorithm for relaxed controls involving a class of hyperbolic systems (1984)
    Springer
    Journal of optimization theory and applications 42 (1984), S. 467-485 
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    ISSN: 1573-2878
    Keywords: Optimal control ; hyperbolic partial differential equations ; Darboux boundary conditions ; strong variational algorithm ; convergence ; relaxed control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we consider a class of optimal control problems involving linear hyperbolic partial differential equations with Darboux boundary conditions. A strong variational algorithm has been obtained for solving this class of optimal control problems in a previous paper by the third and the first authors. It was also shown that anyL ∞ accumulation points of control sequences generated by the algorithm satisfy a necessary condition for optimality. Since such accumulation points need not exist, it is shown in this paper that the control sequences generated by the algorithm always have accumulation points in the sense of control measure, and these accumulation points satisfy a necessary condition for optimality for the corresponding relaxed control problems.
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    URL: http://dx.doi.org/10.1007/BF00935327
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  • 26
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    Necessary and sufficient conditions for optimal control of quasilinear partial differential systems (1984)
    Springer
    Journal of optimization theory and applications 43 (1984), S. 89-101 
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    ISSN: 1573-2878
    Keywords: Optimal control ; distributed-parameter systems ; control of partial differential equations ; sufficiency conditions ; Green's formula
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract First-order necessary and sufficient conditions are obtained for the following quasilinear distributed-parameter optimal control problem: $$max\left\{ {J(u) = \int_\Omega {F(x,u,t) d\omega + } \int_{\partial \Omega } {G(x,t) \cdot d\sigma } } \right\},$$ subject to the partial differential equation $$A(t)x = f(x,u,t),$$ wheret,u,G are vectors andx,F are scalars. Use is made of then-dimensional Green's theorem and the adjoint problem of the equation. The second integral in the objective function is a generalized surface integral. Use of then-dimensional Green's theorem allows simple generalization of single-parameter methods. Sufficiency is proved under a concavity assumption for the maximized Hamiltonian $$H^\circ (x,\lambda ,t) = \max \{ H(x,u,\lambda ,t):u\varepsilon K\} $$ .
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    URL: http://dx.doi.org/10.1007/BF00934748
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    Comparison of a special-purpose algorithm with general-purpose algorithms for solving geometric programming problems (1984)
    Springer
    Journal of optimization theory and applications 43 (1984), S. 237-263 
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    ISSN: 1573-2878
    Keywords: Geometric programming ; computational comparisons ; nonlinear programming ; ellipsoid algorithm ; generalized reduced gradient algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study the performance of four general-purpose nonlinear programming algorithms and one special-purpose geometric programming algorithm when used to solve geometric programming problems. Experiments are reported which show that the special-purpose algorithm GGP often finds approximate solutions more quickly than the general-purpose algorithm GRG2, but is usually not significantly more efficient than GRG2 when greater accuracy is required. However, for some of the most difficult test problems attempted, GGP was dramatically superior to all of the other algorithms. The other algorithms are usually not as efficient as GGP or GRG2. The ellipsoid algorithm is most robust.
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    URL: http://dx.doi.org/10.1007/BF00936164
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    On the gradient-projection method for solving the nonsymmetric linear complementarity problem (1984)
    Springer
    Journal of optimization theory and applications 43 (1984), S. 527-541 
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    ISSN: 1573-2878
    Keywords: Linear complementarity ; nonlinear programming ; gradient projection method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The Levitin-Poljak gradient-projection method is applied to solve the linear complementarity problem with a nonsymmetric matrixM, which is either a positive-semidefinite matrix or aP-matrix. Further-more, if the quadratic functionx T(Mx + q) is pseudoconvex on the feasible region {x ∈R n |Mx + q ≥ 0,x≥0}, then the gradient-projection method generates a sequence converging to a solution, provided that the problem has a solution. For the case when the matrixM is aP-matrix and the solution is nondegenerate, the gradient-projection method is finite.
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    URL: http://dx.doi.org/10.1007/BF00935004
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    Hierarchical multi-objective decision systems for general resource allocation problems (1981)
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    Journal of optimization theory and applications 35 (1981), S. 517-533 
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    ISSN: 1573-2878
    Keywords: Two-level planning ; multi-objective systems ; decentralized systems ; resource allocation ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider optimization methods for hierarchical power-decentralized systems composed of a coordinating central system and plural semi-autonomous local systems in the lower level, each of which possesses a decision making unit. Such a decentralized system where both central and local systems possess their own objective function and decision variables is a multi-objective system. The central system allocates resources so as to optimize its own objective, while the local systems optimize their own objectives using the given resources. The lower level composes a multi-objective programming problem, where local decision makers minimize a vector objective function in cooperation. Thus, the lower level generates a set of noninferior solutions, parametric with respect to the given resources. The central decision maker, then, parametric with respect to the given resources. The central decision maker, then, chooses an optimal resource allocation and the best corresponding noninferior solution from among a set of resource-parametric noninferior solutions. A computational method is obtained based on parametric nonlinear mathematical programming using directional derivatives. This paper is concerned with a combined theory for the multi-objective decision problem and the general resource allocation problem.
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    URL: http://dx.doi.org/10.1007/BF00934930
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    Optimal control of a rotary crane (1981)
    Springer
    Journal of optimization theory and applications 35 (1981), S. 535-557 
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    ISSN: 1573-2878
    Keywords: Optimal control ; rotary crane ; nonlinear systems ; computational algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is concerned with the optimal control of a rotary crane, which makes two kinds of motion (rotation and hoisting) at the same time. The optimal control which transfers a load to a desired place as fast as possible and minimizes the swing of the load during the transfer, as well as the swing at the end of transfer, is calculated on the basis of a dynamic model. A new computational technique is employed for computing the optimal control, and several numerical results are presented.
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    URL: http://dx.doi.org/10.1007/BF00934931
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    Optimal control of distributed nuclear reactors using functional analysis (1981)
    Springer
    Journal of optimization theory and applications 34 (1981), S. 445-458 
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    ISSN: 1573-2878
    Keywords: Optimal control ; distributed parameter systems ; nuclear reactors ; functional analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The minimum norm formalism of functional analysis is applied to the problem of minimizing a quadratic cost functional that penalizes the control effort and the deviations of the neutron flux distribution throughout the reactor core. The conditions for optimality are derived for a general, linearized, reactor model with a finite number of control rods. These conditions take the form of a coupled and finite set of Fredholm's integral equations of the second kind with nondegenerate kernels. An example is presented in which the homogeneous slab reactor model is considered. A contraction mapping algorithm is proposed to compute the optimal control.
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    URL: http://dx.doi.org/10.1007/BF00934682
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    Sensitivity analysis in geometric programming (1982)
    Springer
    Journal of optimization theory and applications 37 (1982), S. 1-21 
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    ISSN: 1573-2878
    Keywords: Sensitivity analysis ; geometric programming ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A unified approach to computing first, second, or higher-order derivatives of any of the primal and dual variables or multipliers of a geometric programming problem, with respect to any of the problem parameters (term coefficients, exponents, and constraint right-hand sides) is presented. Conditions under which the sensitivity equations possess a unique solution are developed, and ranging results are also derived. The analysis for approximating second and higher-order sensitivity generalizes to any sufficiently smooth nonlinear program.
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    URL: http://dx.doi.org/10.1007/BF00934363
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    Additional necessary conditions for optimal control with state-variable inequality constraints (1982)
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    Journal of optimization theory and applications 38 (1982), S. 241-250 
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    ISSN: 1573-2878
    Keywords: Optimal control ; state constraints ; necessary conditions for optimality
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is shown that, when the set of necessary conditions for an optimal control problem with state-variable inequality constraints given by Bryson, Denham, and Dreyfus is appropriately augmented, it is equivalent to the (different) set of conditions given by Jacobson, Lele, and Speyer. Relationships among the various multipliers are given.
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    Global optimization by controlled random search (1983)
    Springer
    Journal of optimization theory and applications 40 (1983), S. 333-348 
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    ISSN: 1573-2878
    Keywords: Numerical optimization ; global search ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper describes a new version, known as CRS2, of the author's controlled random search procedure for global optimization (CRS). The new procedure is simpler and requires less computer storage than the original version, yet it has a comparable performance. The results of comparative trials of the two procedures, using a set of standard test problems, are given. These test problems are examples of unconstrained optimization. The controlled random search procedure can also be effective in the presence of constraints. The technique of constrained optimization using CRS is illustrated by means of examples taken from the field of electrical engineering.
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    URL: http://dx.doi.org/10.1007/BF00933504
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    Second-order necessary conditions in optimal control: Accessory-problem results without normality conditions (1983)
    Springer
    Journal of optimization theory and applications 41 (1983), S. 75-106 
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    ISSN: 1573-2878
    Keywords: Optimal control ; second-order necessary conditions ; accessory problem ; control constraints ; normality condition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An optimal control problem, which includes restrictions on the controls and equality/inequality constraints on the terminal states, is formulated. Second-order necessary conditions of the accessory-problem type are obtained in the absence of normality conditions. It is shown that the necessary conditions generalize and simplify prior results due to Hestenes (Ref. 5) and Warga (Refs. 6 and 7).
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    Uniqueness conditions for Kuhn-Tucker points on a disk (1984)
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    Journal of optimization theory and applications 44 (1984), S. 701-721 
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    ISSN: 1573-2878
    Keywords: Kuhn-Tucker points ; local and global minima ; nonlinear programming ; Morse functions ; convex transformable programs
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Consider minimizingf onD which is diffeomorphic to a disk. Under a genericity assumption, the number of points onD satisfying the Kuhn-Tucker necessary conditions for minimum is odd. We give conditions which imply that a local minimum is global and a necessary and sufficient condition that a Kuhn-Tucker point is the solution. Convex transformable problems satisfy the latter condition.D may be of full dimension or be embedded on a manifold or it may be given by a system of concave inequalities.
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    URL: http://dx.doi.org/10.1007/BF00938403
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    Optimum load scheduling of nuclear-hydro-thermal power systems (1981)
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    Journal of optimization theory and applications 35 (1981), S. 261-275 
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    ISSN: 1573-2878
    Keywords: Optimal control ; load scheduling ; nuclear-hydro-thermal power systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper considers the problem of short-term optimal operation of nuclear-hydro-thermal electric power systems. The solution is obtained by use of a functional analytic optimization technique that employs the minimum norm formulation. A power system with an arbitrary number of generating stations is considered. The limited flexibility exhibited by the thermal nuclear reactors, when operating in a load-following mode, is accounted for by means of a model of the xenon concentration in their cores. The nonlinear effects induced by trapezoidal water reservoirs and the time delay of the water flow between upstream and downstream hydroplants is taken into consideration as well. A two-level iterative scheme of the feasible type is proposed for implementing the optimal solution.
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    URL: http://dx.doi.org/10.1007/BF00934580
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    A compact updating formula for quasi-Newton minimization algorithms (1982)
    Springer
    Journal of optimization theory and applications 36 (1982), S. 477-494 
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    ISSN: 1573-2878
    Keywords: Unconstrained optimization ; variable-metric methods ; quasi-Newton methods ; numerical algorithms ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Quasi-Newton algorithms minimize a functionF(x),x ∈R n, searching at any iterationk along the directions k=−H kgk, whereg k=∇F(x k) andH k approximates in some sense the inverse Hessian ofF(x) atx k. When the matrixH is updated according to the formulas in Broyden's family and when an exact line search is performed at any iteration, a compact algorithm (free from the Broyden's family parameter) can be conceived in terms of the followingn ×n matrix: $$H{_R} = H - Hgg{^T} H/g{^T} Hg,$$ which can be viewed as an approximating reduced inverse Hessian. In this paper, a new algorithm is proposed which uses at any iteration an (n−1)×(n−1) matrixK related toH R by $$H_R = Q\left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & K \\ \end{array} } \right]Q$$ whereQ is a suitable orthogonaln×n matrix. The updating formula in terms of the matrixK incorporated in this algorithm is only moderately more complicated than the standard updating formulas for variable-metric methods, but, at the same time, it updates at any iteration a positive definite matrixK, instead of a singular matrixH R. Other than the compactness with respect to the algorithms with updating formulas in Broyden's class, a further noticeable feature of the reduced Hessian algorithm is that the downhill condition can be stated in a simple way, and thus efficient line searches may be implemented.
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    URL: http://dx.doi.org/10.1007/BF00940543
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    A class of generalized variable penalty methods for nonlinear programming (1981)
    Springer
    Journal of optimization theory and applications 35 (1981), S. 159-182 
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    ISSN: 1573-2878
    Keywords: Variable penalty methods ; nonlinear programming ; sequential unconstrained minimization technique ; approximations ; Hessian matrix ; penalty methods ; ill-conditioning
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A class of generalized variable penalty formulations for solving nonlinear programming problems is presented. The method poses a sequence of unconstrained optimization problems with mechanisms to control the quality of the approximation for the Hessian matrix, which is expressed in terms of the constraint functions and their first derivatives. The unconstrained problems are solved using a modified Newton's algorithm. The method is particularly applicable to solution techniques where an approximate analysis step has to be used (e.g., constraint approximations, etc.), which often results in the violation of the constraints. The generalized penalty formulation contains two floating parameters, which are used to meet the penalty requirements and to control the errors in the approximation of the Hessian matrix. A third parameter is used to vary the class of standard barrier or quasibarrier functions, forming a branch of the variable penalty formulation. Several possibilities for choosing such floating parameters are discussed. The numerical effectiveness of this algorithm is demonstrated on a relatively large set of test examples.
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    Optimal control of a large turboalternator (1981)
    Springer
    Journal of optimization theory and applications 34 (1981), S. 83-97 
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    ISSN: 1573-2878
    Keywords: Optimal control ; large turboalternators ; torque control ; voltage control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The optimal torque and voltage control for a large turbogenerator is found by using the minimum norm formulation. It should be noted that the model used is highly nonlinear. Numerical results are presented.
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    URL: http://dx.doi.org/10.1007/BF00933358
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    Dynamic programming using singular perturbations (1982)
    Springer
    Journal of optimization theory and applications 38 (1982), S. 221-230 
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    ISSN: 1573-2878
    Keywords: Optimal control ; dynamic programming ; singular perturbations ; system order reduction
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The singular perturbation method is used in dynamic programming to reduce the order and the computational requirements of linear systems composed of slow and fast modes. After the fast modes are separated, a near-optimum solution is computed at two different iteration rates determined by the slow and fast subsystem dynamics. The result is a reduction in the order of the computational requirement of the given system to that of the slow subsystem.
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    URL: http://dx.doi.org/10.1007/BF00934084
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    Optimal control of two interconnected turbogenerators (1983)
    Springer
    Journal of optimization theory and applications 40 (1983), S. 255-291 
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    ISSN: 1573-2878
    Keywords: Optimal control ; two interconnected turbogenerators ; torque control ; voltage control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, the optimal control of a system with two identical interconnected turbogenerators, which are connected to an infinite bus, is considered. Control of the generators is effected through control of field voltages and turbine torques. The alternators are controlled through a linear feedback of the state variables. The feedback parameters are obtained by solving a nonlinear, two-point boundary-value problem. The values obtained for these parameters depend on the strength and duration of the disturbance, since the model is nonlinear, in contrast to the usual feedback control of a linear model. The numerical values used are indicated in the Appendix.
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    URL: http://dx.doi.org/10.1007/BF00933940
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    Feasible direction method for large-scale nonconvex programs: Decomposition approach (1981)
    Springer
    Journal of optimization theory and applications 35 (1981), S. 231-249 
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    ISSN: 1573-2878
    Keywords: Optimal control ; nonconvex programming ; constrained optimization ; feasible direction methods ; dual decomposition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we propose a feasible-direction method for large-scale nonconvex programs, where the gradient projection on a linear subspace defined by the active constraints of the original problem is determined by dual decomposition. Results are extended for dynamical problems which include distributed delays and constraints both in state and control variables. The approach is compared with other feasible-direction approaches, and the method is applied to a power generation problem. Some computational results are included.
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    URL: http://dx.doi.org/10.1007/BF00934578
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