An approximate solution of linear multicriteria control problems through the multicriteria simplex method

This paper deals with multicriteria, linear, continuous optimal control problems through the application of the multicriteria simplex method. Since the system is linear, the state variables at a given time can be expressed in terms of definite integrals. Upon using standard numerical integration formulas, the definite integrals are approximately expressed by means of weighted sums of the integrands. Then, after introducing some suitable auxiliary variables, approximate linear multicriteria programming problems are formulated. Illustrative numerical examples are provided to indicate the efficiency of the proposed method.The author is indebted to Professor Y. Sawaragi of Kyoto University for his constant encouragement. The author also wishes to thank T. Suwa for his cooperation in preparing the programming for the digital computer.     

Generalized quasiconvexities,cone saddle points,and minimax theorem for vector-valued functions

We establish the following theorems: (i) an existence theorem for weak type generalized saddle points; (ii) an existence theorem for strong type generalized saddle points; (iii) a generalized minimax theorem for a vector-valued function. These theorems are generalizations and extensions of the author's recent results. For such extensions, we propose new concepts of convexity and continuity of vector-valued functions, which are weaker than ordinary ones. Some of the proofs are based on a few key observations and also on the Browder coincidence theorem or the Tychonoff fixed-point theorem. Also, the minimax theorem follows from the existence theorem for weak type generalized saddle points. The main spaces with mathematical structures considered are real locally convex spaces and real ordered topological vector spaces.This paper is dedicated to Professor Kensuke Tanaka on his sixtieth birthday.This paper was written when the author was a visitor at the Department of Mathematical Science, Graduate School of Science and Technology, Niigata University, Niigata, Japan. The author is indebted to Prof. K. Tanaka for suggesting this work.The author is very grateful to Prof. P. L. Yu for his useful encouragement and suggestions and to the referees for their valuable suggestions and comments.     

Derived Sets for Weak Multiobjective Optimization Problems with State and Control Variables

The concept of a K-gradient, introduced in Ref. 1 in order to generalize the concept of a derived convex cone defined by Hestenes, is extended to weak multiobjective optimization problems including not only a state variable, but also a control variable. The new concept is employed to state multiplier rules for the local solutions of such dynamic multiobjective optimization problems. An application of these multiplier rules to the local solutions of an abstract multiobjective optimal control problem yields general necessary optimality conditions that can be used to derive concrete maximum principles for multiobjective optimal control problems, e.g., problems described by integral equations with additional functional constraints.     

Necessary conditions for nonlinear suboptimization over the weakly-efficient set

The optimization problem of a nonlinear real function over the weakly-efficient set associated to a nonlinear multi-objective program is examined. Necessary first-order conditions for a suboptimal solution are proposed, assuming the convexity of the multi-objective program. Estimations of the optimal value are established and an algorithm for finding suboptimal solutions is proposed. The optimal value is approximated to any prescribed degree of accuracy using a weakly-efficient suboptimal solution.This work was done while the author was preparing his Ph.D. Thesis at the University of Melbourne, Australia. The author is thankful to Dr. B. D. Craven for his suggestions and helpful discussions and to Professor W. Stadler and the anonymous referees for their helpful comments and corrections.     

Optimality conditions and duality in subdifferentiable multiobjective fractional programming

Fritz John and Kuhn-Tucker necessary and sufficient conditions for a Pareto optimum of a subdifferentiable multiobjective fractional programming problem are derived without recourse to an equivalent convex program or parametric transformation. A dual problem is introduced and, under convexity assumptions, duality theorems are proved. Furthermore, a Lagrange multiplier theorem is established, a vector-valued ratio-type Lagrangian is introduced, and vector-valued saddle-point results are presented.The authors are thankful to the referees and Professor P. L. Yu for their many useful comments and suggestions which have improved the presentation of the paper.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant No. A-5319. The authors are also thankful to the Dean's Office, Faculty of Management, University of Manitoba, for the financial support provided for the third author's visit to the Faculty.     

Control synthesis for discrete time systems with control and state bounds in the presence of disturbances

The problem of control in the presence of unknown but limited disturbance for a discrete-time linear system with polyhedral input and state bounds is investigated. Two problems are considered: that of reaching an assigned target set in the state space; and that of keeping the state in a given region using the available controls. In both cases, a solution is given via linear programming. A computational procedure for the control synthesis is proposed which can be implemented to obtain a feedback control.The author thanks Professor G. Leitmann for his helpful suggestions.     

Sensitivity analysis in multiobjective optimization

Sensitivity analysis in multiobjective optimization is dealt with in this paper. Given a family of parametrized multiobjective optimization problems, the perturbation map is defined as the set-valued map which associates to each parameter value the set of minimal points of the perturbed feasible set in the objective space with respect to a fixed ordering convex cone. The behavior of the perturbation map is analyzed quantitatively by using the concept of contingent derivatives for set-valued maps. Particularly, it is shown that the sensitivity is closely related to the Lagrange multipliers in multiobjective programming.This research was made while the author stayed at the International Institute for Applied Systems Analysis, Laxenburg, Austria.The author would like to thank an anonymous referee for his helpful suggestions; particularly, he pointed out that Proposition 2.2 and Theorem 2.1 are valid also in infinite-dimensional spaces.     

Reciprocal Optimal Control Problems and the Associated Pareto Frontier

It is well known that, if a control is Pareto optimal for a multiobjective optimal control problem, then it satisfies the necessary conditions of an optimal control problem with isoperimetric constraints. We introduce a set of sufficient conditions reversing that implication. Thus, we study some properties of the isoperimetric problems and their applications to the analysis of economic models.Research supported by the Ministry of Education, University, and Research of Italy and by the University of Padova.The authors thank the anonymous referee for suggestions that have added clarity to the exposition of the paper.     

Quasiconcavity and nondominated solutions in multiobjective programming

In the multiobjective programming literature, the concavity of the objectives is usually assumed to be a sufficient condition in seeking Pareto-optimal solutions. This paper investigates nondominated solutions associated with dominance cones via the assumption of the quasiconcavity of the objectives. Necessary as well as sufficient conditions for such quasiconcave multiobjective programming problems are obtained.The author is indebted to one of the referees for detailed constructive comments and suggestions. Thanks are also due to the late Professor Abraham Charnes, University of Texas at Austin, and Professor Zhimin Huang, Adelphi University.     

Unified approach to approximate solutions in games and multiobjective programming

Using a theorem of Tijs, we derive results about approximate solutions for Nash equilibrium theory and for multiobjective problems. We describe conditions under which one can replace an infinite strategy set, an infinite alternative set, or an infinite set of criteria by a finite subset without losing all approximate solutions of the problem under consideration.This work was done during the period when the second author was Visiting Professor of the Italian National Research Council at the Mathematical Department of the University of Pavia, Pavia, Italy.     

Necessary conditions for optimality of singular controls in systems governed by partial differential equations

In this paper, we present a method to obtain necessary conditions for optimality of singular controls in systems governed by partial differential equations (distributed-parameter systems). The method is based on the one developed earlier by the author for singular control problems described by ordinary differential equations. As applications, we consider conditions for optimality of singular controls in a Darboux-Goursat system and in control systems that describe chemical processes.This research was supported in part by the National Science Foundation under Grant No. NSF-MCS-80-02337 at the University of Michigan.The author wishes to express his deep gratitude to Professor L. Cesari for his valuable guidance and constant encouragement during the preparation of this paper.     

Duality for Set-Valued Multiobjective Optimization Problems,Part 1: Mathematical Programming

The duality of multiobjective problems is studied with the help of the apparatus of conjugate set-valued mappings introduced by the author. In this paper (Part 1), a duality theory is developed for set-valued mappings, which is then used to derive dual relations for some general multiobjective optimization problems which include convex programming and optimal control problems. Using this result, in the companion paper (Part 2), duality theorems are proved for multiobjective quasilinear and linear optimal control problems. The theory is applied to get dual relations for some multiobjective optimal control problem.     

Optimal control of continuous-time deterministic systems with incomplete discrete feedback

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.The author is grateful to Professor R. W. Rishel for his support during the preparation of the paper.     

Spline approximations for nonlinear hereditary control systems

A spline-based approximation scheme is discussed for optimal control problems governed by nonlinear nonautonomous delay differential equations. The approximating framework reduces the original control problem to a sequence of optimization problems governed by ordinary differential equations. Convergence proofs, which appeal directly to dissipative-type estimates for the underlying nonlinear operator, are given and numerical findings are summarized.This work was supported in part by the Air Force Office of Scientific Research under Contract No. AFOSR-76-3092D, in part by the National Science Foundation under Grants Nos. NSF-MCS-79-05774-05 and NSF-MCS-82-00883, and in part by the US Army Research Office under Contract No. ARO-DAAG29-79-C-0161. The results reported here are a portion of the author's doctoral dissertation written under the supervision of Professor H. T. Banks, Brown University. The author is indebted to Professor Banks for his many valuable comments and suggestions during the course of this work.Part of this research was completed while the author was a visitor at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, Virginia.     

Conjugate duality for multiobjective composed optimization problems

Given a multiobjective optimization problem with the components of the objective function as well as the constraint functions being composed convex functions, we introduce, by using the Fenchel-Moreau conjugate of the functions involved, a suitable dual problem. Under a standard constraint qualification and some convexity as well as monotonicity conditions we prove the existence of strong duality. Finally, some particular cases of this problem are presented.      

Constraint qualifications in multiobjective optimization problems: Differentiable case

In this paper, we are concerned with a multiobjective optimization problem with inequality constraints. We introduce a constraint qualification and derive the Kuhn-Tucker type necessary conditions for efficiency. Moreover, we give conditions which ensure the constraint qualification.This work was done while the author was visiting the University of California, Berkeley, California.     

Necessary conditions for optimality of singular controls

The present paper is concerned with the study of controls which are singular in the sense of the maximum principle. We obtain necessary conditions for optimality of singular controls in systems governed by ordinary differential equations. A useful feature of the method considered here is that it can be applied to optimal control problems with distributed parameters.this research was supported in part by the National Science Foundation under Grant No. NSF-MCS-80-02337 at the University of Michigan.The author wishes to express his deep gratitude to Professor L. Cesari for his valuable guidance and constant encouragement during the preparation of this paper.     

Second-order optimality principle for singular optimal control problems

Recently published results of Gift (Ref. 1) are concerned with the necessary conditions for singular optimal control problems (in the sense of Pontryagin's minimum principle). However, those results are incorrect. An illustrative counterexample is given here.This paper was written while the author studied at Fudan University. The author wishes to thank Professor X. J. Li for his guidance and encouragement. The research was partially supported by NSF of China and the Chinese State Education Commission NSF.     

Note on the nonexceptionality of Brunovský's counterexample

When dealing with the time-optimal problem for linear control systems, there may be a difference between optimal open-loop control and corresponding synthesized feedback control, since in the latter case one is led to allow for generalized (Filippov) solutions. In this note, it is shown that the set of two-dimensional linear control systems with a convex polyhedron as control domain, which exhibit such paradoxical behavior (completely characterized by Brunovský), has a nonempty interior, in a natural and appropriately defined topology on the space of all such linear control systems.This work was done while the author was visiting the Department of Mechanical Engineering, University of California, Berkeley, California, supported by a fellowship of the Italian CNR (Consiglio Nazionale delle Ricerche). While keeping full responsibility for the present work, the author wishes to thank Professor G. Leitmann for his invaluable suggestions, comments, and encouragement.     

Hierarchical generating method for large-scale multiobjective systems

This paper investigates large-scale multiobjective systems in the context of a general hierarchical generating method which considers the problem of how to find the set of all noninferior solutions by decomposition and coordination. A new, unified framework of the hierarchical generating method is developed by integrating the envelope analysis approach and the duality theory that is used in multiobjective programming. In this scheme, the vector-valued Lagrangian and the duality theorem provide the basis of a decomposition of the overall multiobjective system into several multiobjective subsystems, and the envelope analysis gives an efficient approach to deal with the coordination at a high level. The following decomposition-coordination schemes for different problems are developed: (i) a spatial decomposition and envelope coordination algorithm for large-scale multiobjective static systems; (ii) a temporal decomposition and envelope coordination algorithm for multiobjective dynamic systems; and (iii) a three-level structure algorithm for large-scale multiobjective dynamic systems.This work was supported by NSF Grant No. CEE-82-11606.