Showing papers in "Journal of Optimization Theory and Applications in 1972"
TL;DR: In this paper, the extremal value of the linear program as a function of the parameterizing vector and the set of values of the parametric vector for which the program is feasible were derived using linear programming duality theory.
Abstract: J. F. Benders devised a clever approach for exploiting the structure of mathematical programming problems withcomplicating variables (variables which, when temporarily fixed, render the remaining optimization problem considerably more tractable). For the class of problems specifically considered by Benders, fixing the values of the complicating variables reduces the given problem to an ordinary linear program, parameterized, of course, by the value of the complicating variables vector. The algorithm he proposed for finding the optimal value of this vector employs a cutting-plane approach for building up adequate representations of (i) the extremal value of the linear program as a function of the parameterizing vector and (ii) the set of values of the parameterizing vector for which the linear program is feasible. Linear programming duality theory was employed to derive the natural families ofcuts characterizing these representations, and the parameterized linear program itself is used to generate what are usuallydeepest cuts for building up the representations.
2,133 citations
TL;DR: In this article, a third-order pursuit-evasion game with both pursuers and evaders having the same speed and minimum turn radius is described. And the game of kind is first solved for the barrier or envelope of capturable states, and then solved for optimal controls of the two pursuers as functions of the relative position.
Abstract: This paper describes a third-order pursuit—evasion game in which both players have the same speed and minimum turn radius. The game of kind is first solved for thebarrier or envelope of capturable states. When capture is possible, the game of degree is then solved for the optimal controls of the two players as functions of the relative position. The solution is found to include a universal surface for the pursuer and a dispersal surface for the evader.
122 citations
TL;DR: In this article, two modifications of Hestenes' method of multipliers are presented in order to improve its convergence characteristics, and the improved convergence is achieved by increasing the updating frequency so that the number of iterations in a cycle is shortened to ΔN=1 for the ordinary-gradient algorithm and the modified-quasilinearization algorithm and ΔN =n for the conjugate gradient algorithm.
Abstract: In this paper, the numerical solution of the basic problem of mathematical programming is considered. This is the problem of minimizing a functionf(x) subject to a constraint ϕ(x)=0. Here,f is a scalar,x is ann-vector, and ϕ is aq-vector, withq0 is the penalty constant. Previously, the augmented penalty functionW(x, λ,k) was used by Hestenes in his method of multipliers. In Hestenes' version, the method of multipliers involves cycles, in each of which the multiplier and the penalty constant are held constant. After the minimum of the augmented penalty function is achieved in any given cycle, the multiplier λ is updated, while the penalty constantk is held unchanged. In this paper, two modifications of the method of multipliers are presented in order to improve its convergence characteristics. The improved convergence is achieved by (i) increasing the updating frequency so that the number of iterations in a cycle is shortened to ΔN=1 for the ordinary-gradient algorithm and the modified-quasilinearization algorithm and ΔN=n for the conjugate-gradient algorithm, (ii) imbedding Hestenes' updating rule for the multiplier λ into a one-parameter family and determining the scalar parameter β so that the error in the optimum condition is minimized, and (iii) updating the penalty constantk so as to cause some desirable effect in the ordinary-gradient algorithm, the conjugate-gradient algorithm, and the modified-quasilinearization algorithm. For the sake of identification, Hestenes' method of multipliers is called Method MM-1, the modification including (i) and (ii) is called Method MM-2, and the modification including (i), (ii), (iii) is called Method MM-3. Evaluation of the theory is accomplished with seven numerical examples. The first example pertains to a quadratic function subject to linear constraints. The remaining examples pertain to non-quadratic functions subject to nonlinear constraints. Each example is solved with the ordinary-gradient algorithm, the conjugate-gradient algorithm, and the modified-quasilinearization algorithm, which are employed in conjunction with Methods MM-1, MM-2, and MM-3. The numerical results show that (a) for given penalty constantk, Method MM-2 generally exhibits faster convergence than Method MM-1, (b) in both Methods MM-1 and MM-2, the number of iterations for convergence has a minimum with respect tok, and (c) the number of iterations for convergence of Method MM-3 is close to the minimum with respect tok of the number of iterations for convergence of Method MM-2. In this light, Method MM-3 has very desirable characteristics.
100 citations
TL;DR: In this article, a method of computing solutions of variational inequalities in a finite-dimensional space was proposed. But this method is quite close to the method of Theil-Van de Panne described in Ref. 1 in the case of quadratic programming.
Abstract: We observe that variational inequalities generalize convex programming. We look here for a method of computing solutions of variational inequalities in a finite-dimensional space. The method we propose is quite close to the method of Theil-Van de Panne described in Ref. 1 in the case of quadratic programming.
88 citations
TL;DR: In this article, the necessary and sufficient conditions for a group of algorithms to form part of one of these subsets are given, which is the subset of algorithms that also generate sequences of identical points on more general functions.
Abstract: Huang (Ref. 1) introduced a general family of variable metric updating formulas and showed that, for a convex quadratic function, all members of this family generate the same sequence of points and converge in at mostn steps. Huang and Levy (Ref. 2) published numerical data showing the behavior of this family for nonquadratic functions and concluded that this family could be divided into subsets that also generate sequences of identical points on more general functions. In this paper, the necessary and sufficient conditions for a group of algorithms to form part of one of these subsets are given.
85 citations
TL;DR: In this paper, a third-order pursuit-evasion game in which both players have the same speed and minimum turn radius is described. The game of kind is first solved for the barrier or envelope of capturable envelopes.
Abstract: This paper describes a third-order pursuit--evasion game in which both players have the same speed and minimum turn radius. The game of kind is first solved for thebarrier or envelope of capturable...
63 citations
TL;DR: In this paper, it was shown that strong and weak reachability are equivalent to finite-time reachability and controllability for continuous systems in Hilbert space, and that strong reachability is equivalent to the Cayley transform.
Abstract: The notions of reachability and controllability generalize to infinite-dimensional systems in two different ways. We show that the strong notions are equivalent to finite-time reachability and controllability. For discrete systems in Hilbert space, we get simple relations generalizing the Kalman conditions. In the case of a continuous system in Hilbert space, weak reachability is equivalent to the weak reachability of a related discrete system via the Cayley transform.
49 citations
TL;DR: In this article, the explicit feedback control law for the singular linear quadratic-gaussian stochastic control problem is derived and the interesting implication of the control law in terms of information pattern is discussed.
Abstract: The explicit feedback control law for the singular linear-quadratic-gaussian stochastic control problem is derived. The interesting implication of the control law in terms of information pattern is discussed.
38 citations
TL;DR: In this article, a clarification of various items in Ref. 1 is dealt with and additional results concerning control space properties of cooperative games are also presented, where the authors deal with the clarification of several items in ref. 1.
Abstract: This note deals with the clarification of various items in Ref. 1. Additional results concerning control space properties of cooperative games are also presented.
35 citations
TL;DR: The modified multiplier method with the simplified conjugate gradient method is used to compute the solution of a time-optimal control problem for a V/STOL aircraft.
Abstract: A modified multiplier method for optimization problems with equality constraints is suggested and its application to constrained optimal control problems described. For optimal control problems with free terminal time, a gradient descent technique for updating control functions as well as the terminal time is developed. The modified multiplier method with the simplified conjugate gradient method is used to compute the solution of a time-optimal control problem for a V/STOL aircraft.
29 citations
TL;DR: In this paper, a generalization of Farkas' lemma to nonlinear functions and to infinite-dimensional spaces is presented, and the necessary condition for constrained minima is deduced for infinite dimension and cone constraints.
Abstract: Farkas' lemma is generalized both to nonlinear functions and to infinite-dimensional spaces; the version for linear maps is less restricted than Hurwicz's result. A generalization of F. John's necessary condition for constrained minima is deduced for infinite dimension and cone constraints. Some theorems on converse and symmetric duality in nonlinear programming are obtained, which extend the known results, even in the finite-dimensional case.
TL;DR: In this paper, necessary and sufficient conditions for the observability of systems described by nonlinear ordinary differential equations with nonlinear observations are presented, based on extension of the necessary and necessary conditions for observability for time-varying linear systems to the linearized trajectory of the nonlinear system.
Abstract: New necessary and sufficient conditions are presented for the observability of systems described by nonlinear ordinary differential equations with nonlinear observations. The conditions are based on extension of the necessary and sufficient conditions for observability of time-varying linear systems to the linearized trajectory of the nonlinear system. The result is that the local observability of any initial condition can be readily determined, and the observability of the entire initial domain can be computed. The observability of constant parameters appearing in the differential equations is also considered. Examples are presented to illustrate the theory.
TL;DR: An extension of geometric programming to includegeneralized polynomials, and not onlyPositive polynomial, is described in this paper, together with an algorithm and a numerical example.
Abstract: An extension of geometric programming to includegeneralized polynomials, and not onlyPositive polynomials, is described, together with an algorithm and a numerical example.
TL;DR: The results indicate that the inclusion of a restoration phase is necessary for rapid convergence and while SGRA-CR is the most desirable algorithm if feasibility of the suboptimal solutions is required, rapidity of convergence to the optimal solution can be increased if one employs algorithms with incomplete restoration, in particular, CGRA-IR.
Abstract: This paper considers the problem of minimizing a functionalI which depends on the statex(t), the controlu(t), and the parameter π Here,I is a scalar,x ann-vector,u anm-vector, and π ap-vector At the initial point, the state is prescribed At the final point, the state and the parameter are required to satisfyq scalar relations Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential equations
TL;DR: In this article, the authors established sufficient conditions for complete controllability of systems of the form\(dot x = A(t)x + g(t, u), u), where u is the inverse of the fundamental matrix solution of the homogeneous equation.
Abstract: In this paper, we establish sufficient conditions for complete controllability of systems of the form\(\dot x = A(t)x + g(t, u)\). We assume that the inverse of the fundamental matrix solution of the homogeneous equationz=A(t)z is uniformly bounded on [t0, ∞). The criterion used is a growth condition similar to one used by LaSalle on linear control systems. Our results extend his concept of an asymptotically proper control system.
TL;DR: In this paper, a minimax terminal state estimation problem is posed for a linear plant and a generalized quadratic loss function, and sufficient conditions are developed to insure that a Kalman filter will provide a minimum estimate for the terminal state of the plant.
Abstract: A minimax terminal state estimation problem is posed for a linear plant and a generalized quadratic loss function. Sufficient conditions are developed to insure that a Kalman filter will provide a minimax estimate for the terminal state of the plant. It is further shown that this Kalman filter will not generally be a minimax estimate for the terminal state if the observation interval is arbitrarily long. Consequently, a subminimax estimate is defined, subject to a particular existence condition. This subminimax estimate is related to the Kalman filter, and it may provide a useful estimate for the terminal state when the performance of the Kalman filter is no longer satisfactory.
TL;DR: In this article, a minimum principle for the Verhulst-Pearl population equation in terms of the integral of the inverse of population is given, which is the same as the minimum principle given in this paper.
Abstract: A minimum principle for the Verhulst-Pearl population equation is given in terms of the integral of the inverse of population. This principle is contrasted with one given earlier by Volterra in terms of the integral of population.
TL;DR: In this article, the first results obtained in the application of stochastic control theory to flight control problems are presented. But no practical implementation is reported, the implications for such implementation appear to be promising.
Abstract: This paper presents the first results obtained in the application of stochastic control theory to flight control problems. It involves identification and adaptive control of an aircraft which operates over a wide range of environmental conditions that affect its dynamic characteristics. The bulk of the paper deals with theidentification problem of estimating stability derivatives in the presence of turbulence. Simulation results are presented both for identification and control (windgust alleviation and desired response to pilot input). While no practical implementation is reported, the implications for such implementation appear to be promising.
TL;DR: In this paper, the utility of analyzing Pareto-optimal solutions of cooperative differential games with convex control sets was investigated and two criteria were given for the utility and utility of analysis of such games.
Abstract: Cooperative differential games having convex control sets are investigated. Two criteria are given for the utility of analyzing Pareto-optimal solutions of such games. The criteria are shown to determine whether a Pareto-optimal control policy belongs to the boundary of the control set or to the interior. A condition which is necessary for Pareto-optimality in such games is employed to obtain the results. An example is given to demonstrate that the results do not necessarily apply to games having nonconvex control sets.
TL;DR: In this article, Hestenes' method of multipliers is used to approximate a quadratic optimal control problem and the global existence of a family of unconstrained problems is established.
Abstract: Hestenes' method of multipliers is used to approximate a quadratic optimal control problem. The global existence of a family of unconstrained problems is established. Given an initial estimate of the Lagrange multipliers, a convergent sequence of arcs is generated. They are minimizing with respect to members of the above family, and their limit is the solution to the original differentially constrained problem.
TL;DR: In this paper, the authors examined the primer vector which governs optimal solutions for orbital transfer when the central force field has a more general form than the usual inverse-square-force law along a null-thrust are that connects two successive impulses.
Abstract: This paper examines the primer vector which governs optimal solutions for orbital transfer when the central force field has a more general form than the usual inverse-square-force law Along a null-thrust are that connects two successive impulses, the two sets of state and adjoint equations are decoupled This allows the reduction of the problem to the integration of a linear first-order differential equation, and hence the solution of the optimal coasting are in the most general central force field can be obtained by simple quadratures Immediate applications of the results can be seen in solving problems of escape in the equatorial plane of an oblate planet, satellite swing by, or station keeping around Lagrangian points in the three-body problem
TL;DR: In this article, the optimal control is determined for a class of systems by assuming the configuration of the feedback loop, and the gain matrices of the estimator and the controller are so determined that the mean-squared estimation error and the average value of a quadratic cost functional, respectively, are minimized.
Abstract: The optimal control is determined for a class of systems by assuming the configuration of the feedback loop. The feedback loop consists of an unbiased estimator and controller. The gain matrices of the estimator and the controller are so determined that the mean-squared estimation error and the average value of a quadratic cost functional, respectively, are minimized. This is accomplished by the application of the matrix maximum principle to a distributed parameter system. The results indicate that the optimal estimation and the optimal control can be computed independently (separation principle).
TL;DR: In this paper, a suitable sufficiency theorem is applied to obtain minimizing arcs for a family of unconstrained problems, given an initial estimate of the Lagrange multipliers, a convergent sequence of arcs is generated.
Abstract: Hestenes' method of multipliers is used to approximate the classical isoperimetric problem. A suitable sufficiency theorem is first applied to obtain minimizing arcs for a family of unconstrained problems. Given an initial estimate of the Lagrange multipliers, a convergent sequence of arcs is generated. They are minimizing with respect to members of the above family, and their limit is the solution to the original isoperimetric problem.
TL;DR: In this paper, explicit optimality conditions for minimum-weight design of elastic sandwich beams with segmentwise constant structural stiffness, subject to displacement and mean-square stress constraints, are obtained and an iterative procedure that combines the use of the optimality condition with finite-element analysis is proposed and is illustrated by numerical examples.
Abstract: Explicit optimality conditions for minimum-weight design of elastic sandwich beams with segmentwise constant structural stiffness, subject to displacement and mean-square stress constraints, are obtained. An iterative procedure that combines the use of the optimality conditions with finite-element analysis is proposed and is illustrated by numerical examples. These examples suggest that very few iterations are necessary to obtain a good approximation to the optimal design. It is shown that, for practical purposes, the optimization problem may be simplified by using the optimality conditions derived for statically determinate beams instead of those valid for statically indeterminate beams.
TL;DR: In this paper, the problem of minimizing a vector-valued objective function is considered, and a mean-square strategy is derived for minimizing the objective function with respect to a vector.
Abstract: The problem of minimizing a vector-valued objective function is considered, and a mean-square strategy is derived.
TL;DR: In this article, a scale of concepts is introduced which is intermediate between Kuratovski's upper semicontinuity and Cesari's, and examples of problems of control with distributed and boundary parameters are given showing the relevance of the same concepts.
Abstract: In existence theorems for optimal solutions in control theory, concepts of upper semicontinuity of variable sets in Euclidean spaces have been used. In the present paper, a scale of concepts is introduced which is intermediate between Kuratovski's upper semicontinuity and Cesari's upper semicontinuity. First, the relationships and properties of the concepts are studied. Then, examples of problems of control with distributed and boundary parameters are given showing the relevance of the same concepts.
TL;DR: In this paper, it was established that a recently developed initial value method for solving a family of Fredholm integral equations is equivalent to an initial-value version of a classical method based on solving a set of linear equations.
Abstract: It is established that a recently developed initial-value method for solving a family of Fredholm integral equations is equivalent to an initial-value version of a classical method based on solving a set of linear equations. This enables the establishment of the convergence of the numerical method.
TL;DR: In this article, an analytical technique for optimizing the transient response characteristics for a class of passively stabilized satellites in the presence of practical design parameter constraints is presented. But this technique is directed at linearized models and is treated in detail only for quartic characteristic equations.
Abstract: The principal result of this paper is an analytical technique for optimizing the transient response characteristics for a class of passively stabilized satellites in the presence of practical design parameter constraints. The technique developed is directed at linearized models and is treated in detail only for quartic characteristic equations. The results, even for the special case treated, are sufficiently broad to provide a generalization of some techniques which have appeared in the literature. Briefly, it is shown that certain coalescent root configuration represent a hierarchy of transient optima, and conditions are derived which give the optimum transient characteristics as functions of the available variable system parameters.
TL;DR: In this article, a method of conjugate directions, the projection method, for solving unconstrained minimization problems is presented, under the assumption of uniform strict convexity.
Abstract: A method of conjugate directions, the projection method, for solving unconstrained minimization problems is presented. Under the assumption of uniform strict convexity, the method is shown to converge to the global minimizer of the unconstrained problem and to have an (n − 1)-step superlinear rate of convergence. With a Lipschitz condition on the second derivatives, the rate of convergence is shown to be a modifiedn-step quadratic one.