Optimal control of systems with multiple criteria when disturbances are present

Optimal control problems with a vector performance index and uncertainty in the state equations are investigated. Nature chooses the uncertainty, subject to magnitude bounds. For these problems, a definition of optimality is presented. This definition reduces to that of a minimax control in the case of a scalar cost and to Pareto optimality when there is no uncertainty or disturbance present. Sufficient conditions for a control to satisfy this definition of optimality are derived. These conditions are in terms of a related two-player zero-sum differential game and suggest a technique for determining the optimal control. The results are illustrated with an example.This research was supported by AFOSR under Grant No. 76-2923.     

Proper equality constraints and maximization of index vectors

In solving many practical problems, we have to deal with conflictive multiple objectives (in performance, cost, gain, or payoff, etc). Can all such objectives be achieved simultaneously? The general answer is negative. That is, most multiple-objective problems do not have supreme solutions that can satisfy all of the objectives. Many broader definitions of optimality like Pareto optimum, efficient point, noninferior point, etc, have been introduced in various contexts, so that most multiple-objective problems can have optimal solutions. But such optimal solutions do not in general yield unique vectors of optimal indexes of the conflictive multiple objectives. In most cases, we have to make appropriate tradeoffs, compromises, or choices, among those optimal solutions. To obtain the set of all such optimal solutions (in particular, the set of all optimal index vectors), say for a comprehensive study on appropriate tradeoffs, compromises, or choices, a usual practice is to optimize linear combinations of the multiple-objective functions for various weights. The success of such approach relies heavily on a certain directional convexity condition; in other words, if such convexity is absent, this method will fail to obtain essential subsets. The method of proper equality constraints (PEC), however, relies on no convexity condition at all, and through it we can obtain the entire set. In this paper, we attempt to lay the foundation for the method of PEC. We are mainly concerned with obtaining the set of all maximal index vectors, for most of the broader-sense optimal solutions are actually expressed in terms of maximal index vectors (Ref. 1). First, we introduce the notion of quasisupremal vector as a substantially equivalent substitute for, but a rather practical and useful extension of, the notion of maximal vector. Then, we propose and develop the method of PEC for computing the set of all quasisupremal (or maximal) index vectors. An illustrative example in the allocation of funds is given. One of the important conclusions is that optimizing the index of one objective with the indexes of all other objectives equated to some arbitrary constants may still result in inferior solutions. The sensitivity to variations in these constants are examined, and various tests for quasisupremality (maximality, or optimality) are derived in this paper.     

A necessary and sufficient condition for Pareto-optimal security strategies in multicriteria matrix games

In this paper, a scalar game is derived from a zero-sum multicriteria matrix game, and it is proved that the solution of the new game with strictly positive scalarization is a necessary and sufficient condition for a strategy to be a Pareto-optimal security strategy (POSS) for one of the players in the original game. This is done by proving that a certain set, which is the extension of the set of security level vectors in the criterion function space, is convex and polyhedral. It is also established that only a finite number of scalarizations are necessary to obtain all the POSS for a player. An example is included to illustrate the main steps in the proof.This work was done while the author was a Research Associate in the Department of Electrical Engineering at the Indian Institute of Science and was financially supported by the Council of Scientific and Industrial Research, Delhi, India.The author wishes to express his gratefulness to Professor U. R. Prasad for helpful discussions and to two anonymous referees for suggestions which led to an improved presentation.     

Generating pareto-optimal alternatives by a nonfeasible hierarchical method

A hierarchical algorithm for generating Pareto-optimal alternatives for convex multicriteria problems is derived. At the upper level, values for Lagrange multipliers of the coupling constraints are first given. Then at the subsystems, Pareto-optimal values are determined for the subsystem objectives, whereby an additional term or an additional objective is included due to the Lagrange multipliers. In the subsystem optimizations, the coupling equations between the subsystems are not satisfied; therefore, the method is called nonfeasible. Finally, the upper level checks which of the subsystem solutions satisfy the coupling constraints; these solutions are Pareto-optimal solutions for the overall system.     

Preferred coalitions in cooperative differential games

There are many interesting situations which can be described by anN-person general-sum differential game. Such games are characterized by the fact that the strategy of each player depends upon reasonable assumptions about the strategies of the remaining players; and, thus, these games cannot be considered asN uncoupled optimal control problems. In such cases, we say that the game is not strictly competitive, but involves a mutual interest which makes it possible for all of the players to reduce their costs by cooperating with one another, provided the resulting agreement can be enforced. When cooperation is allowed and there are more than two players, there is always the question of whether all possible subcoalitions will be formed with equal ease. This work considers the situation in which a particular subcoalition is preferred. A theory of general-sum games with preferred coalitions is presented, together with constructive examples of alternative approaches which are unsatisfactory.     

Solution concepts in two-person multicriteria games

In this paper, we propose new solution concepts for multicriteria games and compare them with existing ones. The general setting is that of two-person finite games in normal form (matrix games) with pure and mixed strategy sets for the players. The notions of efficiency (Pareto optimality), security levels, and response strategies have all been used in defining solutions ranging from equilibrium points to Pareto saddle points. Methods for obtaining strategies that yield Pareto security levels to the players or Pareto saddle points to the game, when they exist, are presented. Finally, we study games with more than two qualitative outcomes such as combat games. Using the notion of guaranteed outcomes, we obtain saddle-point solutions in mixed strategies for a number of cases. Examples illustrating the concepts, methods, and solutions are included.     

Proper inequality constraints and maximization of index vectors

For obtaining the set of all quasi-supremal index vectors (or all maximal index vectors, or all Pareto-optimal solutions) of a multiple-objective optimization problem, we present, in this paper, the method of proper inequality constraints, which does not rely on any convexity condition at all, but by which one can obtain the entire desired set. This method is based on the observation that optimizing the index of one of the objectives, with some arbitrary bounds assigned to all other objectives, may still result in inferior solutions, unless these bounds areproper. Various necessary and/or sufficient conditions are presented for the properness test.This work was supported by the National Science Foundation under Grant No. GK-32701.     

Maximal vectors and multi-objective optimization

Maximal vector andweak-maximal vector are the two basic notions underlying the various broader definitions (like efficiency, admissibility, vector maximum, noninferiority, Pareto's optimum, etc.) for optimal solutions of multi-objective optimization problems. Moreover, the understanding and characterization of maximal and weak-maximal vectors on the space of index vectors (vectors of values of the multiple objective functions) is fundamental and useful to the understanding and characterization of Pareto-optimal and weak-optimal solutions on the space of solutions.This paper is concerned with various characterizations of maximal and weak-maximal vectors in a general subset of the EuclideanN-space, and with necessary conditions for Pareto-optimal and weak-optimal solutions to a generalN-objective optimization problem having inequality, equality, and open-set constraints on then-space. A geometric method is described; the validity of scalarization by linear combination is studied, and weak conditioning by directional convexity is considered; local properties and a fundamental necessary condition are given. A necessary and sufficient condition for maximal vectors in a simplex or a polyhedral cone is derived. Necessary conditions for Pareto-optimal and weak-optimal solutions are given in terms of Lagrange multipliers, linearly independent gradients, Jacobian and Gramian matrices, and Jacobian determinants.Several advantages in approaching the multi-objective optimization problem in two steps (investigate optimal index vectors on the space of index vectors first, and study optimal solutions on the specific space of solutions next) are demonstrated in this paper.This work was supported by the National Science Foundation under Grant No. GK-32701.     

On a class of three-player differential games

A class of differential games consisting of three players is presented. It is assumed that two of them, forming a coalition and working together, oppose the other. Under some additional assumptions, an optimality criterion for the players who form a coalition is proposed.     

A semistrong sufficiency condition for optimality in nonconvex programming and its connection to the perturbation problem

What happens when a nonconvex program, having a local solutionx ⁰ at which the gradients of the binding constraints are linearly independent, but without strict complementarity hypothesis, is perturbed? Under a relatively weak second-order assumption (some nonnegative second-order terms are supposed to be strictly positive), the perturbed problem has, in the neighborhood ofx ⁰, a finite number of local minima, situated on curves that are connected to some pseudo-solutions of the tangent quadratic problem.     

Sufficient conditions for preference optimality

Preference optimality is an optimality concept in multicriteria problems, that is, in problems where several criteria are to beoptimized simultaneously. Formally, one wishes to optimizeN criteriag _i(·) or, equivalently, a criterion vectorg(·) ^N , subject to either functional constraints in programming or to side conditions which are differential equations in optimal control. Subject to these constraints, one obtains forg(·) a set of attainable values in ^N . This set is preordered by the introduction of a complete preordering ; a controlu*(·) or a decisionx*, then, is preference-optimal if it results ing(u*(·))g(u(·)) for all admissible controlsu(·) or ifg(x*)g(x) for all feasible decisionsx. The present paper concerns sufficient conditions for preference-optimal control and for preference-optimal decisions.     

A sufficient condition for the evadability of differential evasion games

A sufficient condition for the strict evadability of nonlinear differential evasion games is obtained. The result complements, in some sense, the relevant results obtained by the author in a previous paper. An illustrative example is discussed as well. The author thanks Professor L. D. Berkovitz for some discussions.     

Equilibrium solutions for multiobjective bimatrix games incorporating fuzzy goals

Equilibrium solutions in terms of the degree of attainment of a fuzzy goal for games in fuzzy and multiobjective environments are examined. We introduce a fuzzy goal for a payoff in order to incorporate ambiguity of human judgments and assume that a player tries to maximize his degree of attainment of the fuzzy goal. A fuzzy goal for a payoff and the equilibrium solution with respect to the degree of attainment of a fuzzy goal are defined. Two basic methods, one by weighting coefficients and the other by a minimum component, are employed to aggregate multiple fuzzy goals. When the membership functions are linear, computational methods for the equilibrium solutions are developed. It is shown that the equilibrium solutions are equal to the optimal solutions of mathematical programming problems in both cases. The relations between the equilibrium solutions for multiobjective bimatrix games incorporating fuzzy goals and the Pareto-optimal equilibrium solutions are considered.     

A note on nonzero-sum differential games with bargaining solution

The extension of Nash's bargaining solution to differential games is discussed. It is shown that a closed-loop solution verifies very stringent necessary conditions and that an open-loop solution can present serious weakness from a normative point of view.This research has been supported by the Canada Council (S73-0935) and the Ministère de l'Education du Québec (DGES).     

Multicriteria equilibrium programming: Extraproximal methods

Multicriteria equilibrium optimization is an efficient tool for mathematical modeling of various situations in operations research, design automation, control, etc. In this paper, a formal formulation of the problem of multicriteria equilibrium optimization is given, and an approach to solving this problem is examined.     

Differential Conditions for Constrained Nonlinear Programming via Pareto Optimization

We deal with the differential conditions for local optimality. The conditions that we derive for inequality constrained problems do not require constraint qualifications and are the broadest conditions based on only first-order and second-order derivatives. A similar result is proved for equality constrained problems, although the necessary conditions require the regularity of the equality constraints.     

拟阵上合作对策的Banzhaf函数

本文结合文[1,2]中关于拟阵上静态结构和动态结构合作对策Shapley函数的描述,探讨了两类拟阵上的Banzhaf函数.通过给出相应的公理体系,论述了两类拟阵上Banzhaf函数的存在性和唯一性,拓展了拟阵上分配指标的研究范围.同时讨论了两类合作对策上Banzhaf函数的有关性质.最后通过算例来说明局中人在此类合作对策中的Banzhaf指标.     

A fast Pareto genetic algorithm approach for solving expensive multiobjective optimization problems

We present a new multiobjective evolutionary algorithm (MOEA), called fast Pareto genetic algorithm (FastPGA), for the simultaneous optimization of multiple objectives where each solution evaluation is computationally- and/or financially-expensive. This is often the case when there are time or resource constraints involved in finding a solution. FastPGA utilizes a new ranking strategy that utilizes more information about Pareto dominance among solutions and niching relations. New genetic operators are employed to enhance the proposed algorithm’s performance in terms of convergence behavior and computational effort as rapid convergence is of utmost concern and highly desired when solving expensive multiobjective optimization problems (MOPs). Computational results for a number of test problems indicate that FastPGA is a promising approach. FastPGA yields similar performance to that of the improved nondominated sorting genetic algorithm (NSGA-II), a widely-accepted benchmark in the MOEA research community. However, FastPGA outperforms NSGA-II when only a small number of solution evaluations are permitted, as would be the case when solving expensive MOPs.     

A survey of multicriteria optimization or the vector maximum problem,part I: 1776–1960

In this survey, the history of the subject from 1776 until 1960 is presented. A brief biographical sketch of Vilfredo Pareto is given first. Then, the more or less simultaneous development of the concepts of utility, preference, and welfare theory follows, with results which go back to Hausdorff and Cantor. A brief discussion of the work of Borel and von Neumann as initiators of game theory is included. Each of these areas has developed enough to warrant its own survey; hence, they are reviewed here only insofar as they provide necessary foundations. Thereafter, the concepts of efficiency, vector maximum problem, and Pareto optimality are reviewed in connection with production theory, programming, and economics. The survey is presented within a unified mathematical framework, and the emphasis is on mathematical results, rather than psychological or socio-economic discussion. To enable the reader to draw conclusions without having to obtain each article himself, the results have been presented in somewhat more detail than usual.     

On saddle-point optimality in differential games

A family of two-person, zero-sum differential games in which the admissible strategies are Borel measurable is defined, and two types of saddle-point conditions are introduced as optimality criteria. In one, saddle-point candidates are compared at each point of the state space with all playable pairs at that point; and, in the other, they are compared only with strategy pairs playable on the entire state space. As a theorem, these two types of optimality are shown to be equivalent for the defined family of games. Also, it is shown that a certain closure property is sufficient for this equivalence. A game having admissible strategies everywhere constant, in which the two types of saddle-point candidates are not equivalent, is discussed.This paper is based on research supported by ONR.