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  • Articles  (53)
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  • Articles  (53)
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  • Springer  (53)
  • American Chemical Society
  • American Institute of Physics
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  • 2005-2009
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  • 1
    Electronic Resource
    Electronic Resource
    Conjugate direction methods with variable storage (1982)
    Springer
    Mathematical programming 23 (1982), S. 326-340 
    Details
    ISSN: 1436-4646
    Keywords: Optimization ; Quasi-Newton ; Conjugate Gradient
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we study conjugate gradient algorithms for large optimization problems. These methods accelerate (or precondition) the conjugate gradient method by means of quasi-Newton matrices, and are designed to utilize a variable amount of storage, depending on how much information is retained in the quasi-Newton matrices. We are concerned with the behaviour of such methods on the underlying quadratic model, and in particular, with finite termination properties.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01583797
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  • 2
    Electronic Resource
    Electronic Resource
    A smoothing-out technique for min—max optimization (1980)
    Springer
    Mathematical programming 19 (1980), S. 61-77 
    Details
    ISSN: 1436-4646
    Keywords: Optimization ; Nonlinear Programming ; Unconstrained Optimization ; Nondifferentiable Optimization ; Min—Max Problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper, we suggest approximations for smoothing out the kinks caused by the presence of “max” or “min” operators in many non-smooth optimization problems. We concentrate on the continuous-discrete min—max optimization problem. The new approximations replace the original problem in some neighborhoods of the kink points. These neighborhoods can be made arbitrarily small, thus leaving the original objective function unchanged at almost every point ofR n . Furthermore, the maximal possible difference between the optimal values of the approximate problem and the original one, is determined a priori by fixing the value of a single parameter. The approximations introduced preserve properties such as convexity and continuous differentiability provided that each function composing the original problem has the same properties. This enables the use of efficient gradient techniques in the solution process. Some numerical examples are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581628
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  • 3
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    An algorithm for linear programming which requires O(((m+n)n 2+(m+n)1.5 n)L) arithmetic operations (1990)
    Springer
    Mathematical programming 47 (1990), S. 175-201 
    Details
    ISSN: 1436-4646
    Keywords: Optimization ; linear programming ; complexity ; polynomial time algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We present an algorithm for linear programming which requires O(((m+n)n 2+(m+n)1.5 n)L) arithmetic operations wherem is the number of constraints, andn is the number of variables. Each operation is performed to a precision of O(L) bits.L is bounded by the number of bits in the input. The worst-case running time of the algorithm is better than that of Karmarkar's algorithm by a factor of $$\sqrt {m + n} $$ .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580859
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  • 4
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    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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  • 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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    Equivalence theorem for constrained optimization of the LegendreK-transform of a concave function (1982)
    Springer
    Journal of optimization theory and applications 38 (1982), S. 25-31 
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    ISSN: 1573-2878
    Keywords: Optimization ; Kuhn-Tucker theorem ; LegendreK-transform ; equilibrium composition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The most common way of determining the steady states of a system is through the optimization of a concave function on a convex set. This applies only to cases where the objective function depends exclusively onextensive variables. In fields such as econometrics, physics, or chemistry, controllable quantities (and therefore constrained variables) are generally referred to asintensive parameters, and the states are described through a potential function. In the following pages, we examine how these two aspects can be connected.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934320
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  • 7
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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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  • 8
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    An approach to the solution of the vector optimization problem of dynamic systems (1982)
    Springer
    Journal of optimization theory and applications 38 (1982), S. 409-422 
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    ISSN: 1573-2878
    Keywords: Optimization ; scalar optimization ; vector optimization ; optimization under vector-valued criteria ; maximum principle ; dynamic programming ; optimization of dynamic systems ; multi-criteria decision problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The optimal control problem with vector-valued criteria is considered. A new approach to the generalization of this problem and a method of constructing the Bellman function are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00935346
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  • 9
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    Existence of value in differential games with terminal cost function (1983)
    Springer
    Journal of optimization theory and applications 39 (1983), S. 89-104 
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    ISSN: 1573-2878
    Keywords: Differential games ; optimal strategies ; value of a game ; terminal cost function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A two-person, zero-sum differential game with general type phase constraints and terminal (not fixed) cost function is investigated. Player II (possessing complete information) can choose any strategy in the Varaiya-Lin sense, while his opponent (having incomplete information) can select any lower II-strategy introduced by Friedman (Ref. 1). The existence of a value and an optimal player II's strategy is obtained under assumptions ensuring that the sets of all admissible trajectories for the two players are compact in the Banach space of all continuous functions. The present paper largely extends the results of Ref. 2.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934607
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  • 10
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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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  • 11
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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.
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    URL: http://dx.doi.org/10.1007/BF00934590
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  • 12
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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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  • 13
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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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  • 14
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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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  • 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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    Multimodel design of a Nash strategy (1980)
    Springer
    Journal of optimization theory and applications 31 (1980), S. 553-564 
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    ISSN: 1573-2878
    Keywords: Differential games ; perturbation theory ; systems order reduction ; large-scale systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For interconnected systems with slow and fast modes, we analyze a multimodel design of a Nash equilibrium strategy. Decision makers use different models of the same system to obtain a composite strategy that approximates the exact Nash strategy. It is discussed in what sense this approximation holds.
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    URL: http://dx.doi.org/10.1007/BF00934477
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  • 17
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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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  • 18
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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.
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    URL: http://dx.doi.org/10.1007/BF00934844
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  • 19
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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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  • 20
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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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  • 21
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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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  • 22
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    Remarks on capture-avoidance games (1980)
    Springer
    Journal of optimization theory and applications 32 (1980), S. 365-377 
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    ISSN: 1573-2878
    Keywords: Differential games ; pursuit and evasion ; capture avoidance ; linear games
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We formulate the problems of capture and avoidance as differential games with terminal cost, thus avoiding some difficulties arising from choosing the terminal time or the closest distance as the cost. This puts capture and avoidance in the context of differential games, in the sense of Isaacs. The results are illustrated via linear differential games with hard control constraints.
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    System theory: A new approach to wave mechanics (1980)
    Springer
    Journal of optimization theory and applications 32 (1980), S. 463-478 
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    ISSN: 1573-2878
    Keywords: Optimization ; control ; relativity ; stochastic processes ; wave mechanics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The intention of this paper is to show how the influence of the work of Bellman has initiated a tentative new approach to quantum mechanics
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    URL: http://dx.doi.org/10.1007/BF00934034
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  • 24
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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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  • 25
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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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  • 26
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    Shape design sensitivity of a membrane (1983)
    Springer
    Journal of optimization theory and applications 40 (1983), S. 595-623 
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    ISSN: 1573-2878
    Keywords: Optimization ; partial differential equations ; elasticity ; control ; design sensitivity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The dependence of the static response and the eigenvalues of a membrane on its shape is characterized. A transformation function is defined to determine the shape of the membrane. Differential operator properties and transformation techniques of integral calculus are employed to show that the static response and the eigenvalues of the system depend in a continuous and differentiable way on the shape of the membrane. Explicit and computable formulas are presented for the derivative (first variation) of the structural response and the eigenvalues with respect to the shape. A rigorous proof is provided, and the shape design sensitivity of a typical integral functional is determined.
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    URL: http://dx.doi.org/10.1007/BF00933973
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  • 27
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    On qualitative differential games with two targets (1983)
    Springer
    Journal of optimization theory and applications 41 (1983), S. 587-598 
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    ISSN: 1573-2878
    Keywords: Differential games ; first reachability of sets ; playability ; pursuit-evasion problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider a qualitative differential game between two players, each with their own respective target. A player's objective is to steer the response of a dynamical system under the control of both players into his own target before his opponent can steer it to his. The sufficiency conditions given by Getz and Leitmann (Ref. 2), which ensure termination on a particular target, are improved. The theory is illustrated by a pursuit-evasion problem.
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    Optimal control of continuous-time deterministic systems with incomplete discrete feedback (1990)
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    Journal of optimization theory and applications 64 (1990), S. 557-571 
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    ISSN: 1573-2878
    Keywords: Optimal control ; deterministic systems ; discrete incomplete information ; sufficient conditions for optimality ; feedback systems
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    Topics: Mathematics
    Notes: Abstract Sufficient conditions for optimality are obtained for controls that depend on the current time and values of known functions of the state vector at finite points of the time interval. The equations for finding the required control laws are derived. An example is given for which an exact solution of the problem can be obtained.
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    On a student-related optimal control problem (1990)
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    Journal of optimization theory and applications 65 (1990), S. 129-138 
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    ISSN: 1573-2878
    Keywords: Optimal control ; mathematical modelling ; study strategies
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    Topics: Mathematics
    Notes: Abstract In a recent paper (Ref. 1), Cheng and Teo discussed some further extensions of a student-related optimal control problem which was originally proposed by Raggettet al. (Ref. 2) and later on modified by Parlar (Ref. 3). In this paper, we treat further extensions of the problem.
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    On the minimality of search paths under absolutely continuous univariate distributions (1990)
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    Mathematical methods of operations research 34 (1990), S. 43-58 
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    ISSN: 1432-5217
    Keywords: Linear Search ; Optimization ; Absolutely Continuous distributions
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    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung Ein Objekt, das sich in zufälliger Position auf einer Strecke befindet, soll gesucht werden; der Startpunkt liegt im Inneren der Strecke. Für die Bewegungsgeschwindigkeit bei der Suche ist eine obere Schranke gegeben; der Suchpfad muß stetig sein, wobei beliebig viele Richtungsänderungen zugelassen sind. Das Ziel ist die Minimierung der erwarteten Zeit bis zum Erreichen des Objekts. Unter der Voraussetzung einer absolut stetigen Verteilung für die Position des Objekts werden hinreichende Bedingungen für die Optimalität eines Suchpfades angegeben.
    Notes: Abstract The linear search problem is concerned with the search for a target located randomly on a line. The target being sought for might be on either side of some starting point. A searcher makes for the target with an upper bound on his speed, using a continuous search path in which he changes his direction many times before reaching his goal. The minimality of average distance (time) from the starting point upon reaching the target is the measure of optimality of search paths. We have obtained sufficient conditions that guarantee the minimality of such search paths whenever the target has an absolutely continuous distribution.
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    On the isochronous and stroboscopic capture in differential games (1982)
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    Journal of optimization theory and applications 37 (1982), S. 115-119 
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    ISSN: 1573-2878
    Keywords: Differential games ; isochronous capture ; stroboscopic strategy
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    Topics: Mathematics
    Notes: Abstract In Ref. 1, Hájek formulated the conjecture that isochronous capture is necessarily stroboscopic in linear differential games. Here, it is shown that this conjecture is not valid.
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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
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    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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    Multiplier rules and the separation of convex sets (1983)
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    Journal of optimization theory and applications 40 (1983), S. 321-331 
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    ISSN: 1573-2878
    Keywords: Optimization ; necessary conditions ; Kuhn-Tucker theorem ; multiplier rules ; separation of convex sets ; mathematical programming ; generalized derivatives
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    Topics: Mathematics
    Notes: Abstract We consider the following abstract mathematical programming problem: in a setD, find an element that optimizes a real function φ0, subject to inequality constraints φ1⩽0, ..., φ p ⩽0 and equality constraints φ p+1=0, ..., φ p+q =0. Necessary conditions for this problem, like the Karush-Kuhn-Tucker theorem, can be seen as a consequence of separating with a hyperplane two convex sets inR p+q+1, the image space of the map Φ=(φ0, φ1, ..., φ p+q ). This paper reviews this approach and organizes it into a coherent way of looking at necessary conditions in optimization theory.
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    Second-order necessary conditions in optimal control: Accessory-problem results without normality conditions (1983)
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    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
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    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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    A class of differential games for which the closed-loop and open-loop Nash equilibria coincide (1982)
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    Journal of optimization theory and applications 36 (1982), S. 253-262 
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    ISSN: 1573-2878
    Keywords: Differential games ; Nash equilibria ; open-loop strategies ; closed-loop strategies
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    Topics: Mathematics
    Notes: Abstract It is well known that, in general, Nash equilibria in open-loop strategies do not coincide with those in closed-loop strategies. This note identifies a class of differential games in which the Nash equilibrium in closed-loop strategies is degenerate, in the sense that it depends on time only. Consequently, the closed-loop equilibrium is also an equilibrium in open-loop strategies.
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    On the convergence, invariance, and related aspects of a modification of Huang's algorithm (1982)
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    Journal of optimization theory and applications 37 (1982), S. 315-341 
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    ISSN: 1573-2878
    Keywords: Optimization ; unconstrained minimization ; updates ; line searches ; convergence ; numerical methods
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    Topics: Mathematics
    Notes: Abstract We consider a certain generalization of the Huang family of updates and discuss, firstly, convergence, dependence on parameters, and descent property; secondly, invariance under nonlinear scaling, conjugacy of search directions, and possibility of achieving a better approximation of the inverse of the Hessian. The last three aspects are shown to be dependent on particular choices of parameters. A numerical experiment is presented comparing the performances of the usual and modified BFGS algorithms.
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    Discrete approximation of relaxed optimal control problems (1990)
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    Journal of optimization theory and applications 65 (1990), S. 395-407 
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    ISSN: 1573-2878
    Keywords: Optimal control ; nonlinear systems ; discretization ; nonconvexity ; relaxed controls ; approximation
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    Topics: Mathematics
    Notes: Abstract We consider a general nonlinear optimal control problem for systems governed by ordinary differential equations with terminal state constraints. No convexity assumptions are made. The problem, in its so-called relaxed form, is discretized and necessary conditions for discrete relaxed optimality are derived. We then prove that discrete optimality [resp., extremality] in the limit carries over to continuous optimality [resp., extremality]. Finally, we prove that limits of sequences of Gamkrelidze discrete relaxed controls can be approximated by classical controls.
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    Differential game of optimal approach of two inertial pursuers to a noninertial evader (1990)
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    Journal of optimization theory and applications 65 (1990), S. 501-518 
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    ISSN: 1573-2878
    Keywords: Differential games ; value function ; programmed maximin function ; dispersal surface ; focal surface
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    Topics: Mathematics
    Notes: Abstract In this paper, the game of the optimal approach of two identical inertial pursuers to a noninertial evader is investigated. The duration of the game is fixed. The payoff functional is the distance between the evader and the closest pursuer when the game terminates. The value function is constructed for all possible positions of the game. The regions where the pursuit is one-to-one and the regions where it is essentially collective are described algorithmically. Some analogies between this game and the linear differential game with elliptical vectograms are indicated. It is noted that the focal surface and the dispersal surface are in proximity of one another.
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    Optimal production mix (1990)
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    Journal of optimization theory and applications 66 (1990), S. 255-273 
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    ISSN: 1573-2878
    Keywords: Optimal control ; state constraints ; production problems
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    Topics: Mathematics
    Notes: Abstract This paper deals with a production plant in which two different products can be produced. The plant consists of three subsystemsS i . Before or after a phase of separate processing in subsystemsS 1 andS 2, the two products have to be processed in subsystemS 3. Each of these subsystems has a limited capacity. In the first part, we assume empty stocks at the beginning; at a fixed timeT in the future, certain quantitiesX i of the two products have to be delivered to the customers. Facing linear holding costs, convex production costs, and stringent capacity constraints, the problem is to decide when to produce which product at what rate. It is shown that the optimal solution consists of up to six different regimes and that the time paths of the production rates need not be monotonic. These results, which can be obtained analytically, are also illustrated in several numerical examples. Finally, the case is considered where the terminal demand at timeT is replaced by a continuous and seasonally fluctuating demand rate. It is demonstrated that the optimal production rates show an interesting and nontrivial behavior. In particular, it may happen that, on intervals where the demand for the one product increases, the optimal production rate decreases. This is also demonstrated by computer plots in some numerical examples.
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    Small penalty control of the end temperature in a long rod (1990)
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    Journal of optimization theory and applications 66 (1990), S. 443-454 
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    ISSN: 1573-2878
    Keywords: Optimal control ; heat transfer ; singularly perturbed integral equations
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    Topics: Mathematics
    Notes: Abstract The problem considered is that of maintaining the end temperature of a long rod near a prescribed level over a fixed time interval. Control is achieved via the heat flux at the near end, and it is optimal in the sense that it minimizes a given performance index of quadratic form. The performance index contains a penalty parameter associated with the magnitude of the control. Particular attention is given to the determination of the optimal control when the penalty parameter is small (i.e., cheap control). This gives rise to a singularly perturbed integral equation, which is solved asymptotically by a methodology which has recently been developed for a related class of problems.
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    On the optimal control of minimum film thickness (1990)
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    Journal of optimization theory and applications 67 (1990), S. 17-41 
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    ISSN: 1573-2878
    Keywords: Optimal control ; existence ; numerical approximation ; piston rings ; journal bearings ; hydrodynamic lubrication theory
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    Topics: Mathematics
    Notes: Abstract The present paper is concerned with the study of controlling the motion of a bearing so that the thin film of lubricant separating it from its container will have the largest possible thickness. Explicit equations are derived, and the control problem is solved explicitly in some simple cases.
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    Time-axis decomposition of large-scale optimal control problems (1990)
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    Journal of optimization theory and applications 67 (1990), S. 259-277 
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    ISSN: 1573-2878
    Keywords: Optimal control ; time-axis decomposition ; Stackelberg games ; bilevel programming
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    Topics: Mathematics
    Notes: Abstract Continuous-time optimal control problems can rarely be solved directly but have to be approximated with discrete analogues. Shorter time steps lead to more accurate approximations, but result in formulations that are often too big for computer memory. This paper presents a technique for decomposing the problem along the time axis and iterating toward a solution in a leader-follower framework. In the model, the leader controls a set of coordination parameters, which he passes to the followers, who then solve their individual subproblems. State and sensitivity information is returned to the leader, who attempts to minimize an unconstrained problem in the coordination space. Parameters are updated and the process continues until improvement ceases. Two advantages of this technique are that feasible solutions to the original problem are available at each iteration and that the optimal coordination parameters obtained provide some measure of feedback control. Computational results are presented for a comprehensive set of test problems.
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    Optimal control of time-delay systems with distributed parameters (1990)
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    Journal of optimization theory and applications 67 (1990), S. 567-585 
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    ISSN: 1573-2878
    Keywords: Optimal control ; parabolic partial differential equations ; time-delay systems ; necessary conditions ; sufficient conditions ; maximum principle
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    Topics: Mathematics
    Notes: Abstract An optimal control problem with a prescribed performance index for parabolic systems with time delays is investigated. A necessary condition for optimality is formulated and proved in the form of a maximum principle. Under additional conditions, the maximum principle gives sufficient conditions for optimality. It is also shown that the optimal control is unique. As an illustration of the theoretical consideration, an analytic solution is obtained for a time-delayed diffusion system.
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    Singular differential game numerical techniques (1982)
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    Journal of optimization theory and applications 37 (1982), S. 69-87 
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    ISSN: 1573-2878
    Keywords: Differential games ; singular arcs ; saddle points ; two-point boundary-value problems ; switching times ; closed-loop solutions ; inverse systems
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    Topics: Mathematics
    Notes: Abstract This paper develops a numerical technique to solve a class of zero-sum differential games with singular control. By using this technique and the application of inverse systems, a near-optimal closed-loop technique is developed to generate a numerical solution to this class of problems.
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    Remarks on capture-avoidance games (1982)
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    Journal of optimization theory and applications 37 (1982), S. 121-122 
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    ISSN: 1573-2878
    Keywords: Differential games ; pursuit and evasion ; captureavoidance ; linear games
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    Topics: Mathematics
    Notes: Abstract Corrections to some results of Ref. 1 are stated.
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    Dynamic programming using singular perturbations (1982)
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    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
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    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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    Existence of value in differential games with fixed time duration (1982)
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    Journal of optimization theory and applications 38 (1982), S. 581-598 
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    ISSN: 1573-2878
    Keywords: Differential games ; optimal strategies ; value of a game
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    Topics: Mathematics
    Notes: Abstract A differential game of prescribed duration with general-type phase constraints is investigated. The existence of a value in the Varaiya-Lin sense and an optimal strategy for one of the players is obtained under assumptions ensuring that the sets of all admissible trajectories for the two players are compact in the Banach space of all continuous functions. These results are next widened on more general games, examined earlier by Varaiya.
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    Optimal control of two interconnected turbogenerators (1983)
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    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
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    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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    Feedback Nash equilibria in a problem of optimal fishery management (1990)
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    Journal of optimization theory and applications 64 (1990), S. 293-310 
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    ISSN: 1573-2878
    Keywords: Differential games ; feedback Nash equilibria ; renewable resources ; common-property fishery
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    Topics: Mathematics
    Notes: Abstract The paper deals with a problem of optimal management of a common-property fishery, modelled as a two-player differential game. Under nonclassical assumptions on harvest rates and utilities, a feedback Nash equilibrium is determined, using a bionomic equilibrium concept. Later on, this assumption is relaxed and a feedback Nash equilibrium is established under minimal hypotheses.
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    Optimization technique based on learning automata (1990)
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    Journal of optimization theory and applications 64 (1990), S. 331-347 
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    ISSN: 1573-2878
    Keywords: Optimization ; learning systems ; nonconvex programming ; process synthesis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Optimization techniques are finding increasingly numerous applications in process design, in parallel to the increase of computer sophistication. The process synthesis problem can be stated as a largescale constrained optimization problem involving numerous local optima and presenting a nonlinear and nonconvex character. To solve this kind of problem, the classical optimization methods can lead to analytical and numerical difficulties. This paper describes the feasibility of an optimization technique based on learning systems which can take into consideration all the prior information concerning the process to be optimized and improve their behavior with time. This information generally occurs in a very complex analytical, empirical, or know-how form. Computer simulations related to chemical engineering problems (benzene chlorination, distillation sequence) and numerical examples are presented. The results illustrate both the performance and the implementation simplicity of this method.
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    Optimization of control setting and size of shunt capacitors on distribution feeders (1990)
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    Journal of optimization theory and applications 65 (1990), S. 305-319 
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    ISSN: 1573-2878
    Keywords: Optimization ; shunt capacitors ; distribution feeders ; control
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    Topics: Mathematics
    Notes: Abstract Loads on electric utility systems have two components: active power (measured in kilowatts) and reactive power (measured in kilovars). Active power has to be generated at the power plant, whereas reactive power can be provided by either power plants or capacitors. It is a well-known fact that shunt power capacitors are the most economical source to meet the reactive power requirements of inductive loads and transmission lines operating at a lagging power factor. This paper describes new contributions to the problem of optimization of size and control setting of shunt capacitors on distribution feeders, so that the losses along the feeder are minimized. The variation of the KVAR of the load on the feeder with the distance from the substation is assumed to be linear. The parameters of this function are estimated from the available KVAR loading on the feeder first by using least-square techniques and then by using least-absolute-value parameter estimation techniques. The results obtained are compared with that obtained if the current profile is assumed to be uniformly distributed on the feeder. Our results show that the optimum size of the capacitor bank as well as its optimum location depend on the parameters of the model used for the load. Also, our results show a large saving in the size of the capacitor banks used with a considerable per-unit-loss reduction along the feeder.
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    Maximum principle, dynamic programming, and their connection in deterministic control (1990)
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    Journal of optimization theory and applications 65 (1990), S. 363-373 
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    ISSN: 1573-2878
    Keywords: Optimal control ; maximum principle ; dynamic programming ; viscosity solutions ; superdifferential ; subdifferential
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Two major tools for studying optimally controlled systems are Pontryagin's maximum principle and Bellman's dynamic programming, which involve the adjoint function, the Hamiltonian function, and the value function. The relationships among these functions are investigated in this work, in the case of deterministic, finite-dimensional systems, by employing the notions of superdifferential and subdifferential introduced by Crandall and Lions. Our results are essentially non-smooth versions of the classical ones. The connection between the maximum principle and the Hamilton-Jacobi-Bellman equation (in the viscosity sense) is thereby explained by virtue of the above relationship.
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    Optimal control of linear time-varying systems via Fourier series (1990)
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    Journal of optimization theory and applications 65 (1990), S. 375-384 
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    ISSN: 1573-2878
    Keywords: Optimal control ; time-varying systems ; Fourier series operational matrix ; Fourier series approximation
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    Topics: Mathematics
    Notes: Abstract The optimal control of linear time-varying systems with quadratic cost functional is obtained by Fourier series approximation. The properties of Fourier series are first briefly presented and the operational matrix of backward integration together with the product operational matrix are utilized to reduce the optimal control problem to a set of simultaneous linear algebraic equations. An illustrative example is included to demonstrate the validity and applicability of the technique.
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