Zero-sum games with incompletely defined payoff

It is shown that a saddle-point solution exists in a two-person, zero-sum game whose payoff is given by a matrix which is not completely defined. On the other hand, we show that such games do not always have a value, so that a saddle-point solution is not necessarily an optimal solution.This work was supported by the Centre d'Etudes Atomiques, Saclay, France.     

Optimality in matrix games with incompletely defined payoff

We study two-person, zero-sum matrix games whose payoffs are not defined for every pair of strategies. A necessary and sufficient condition for these games to possess a value is given, and we show that the value can be approximated by using universally playable strategies.This work was supported by the Centre d'Etudes Nucléaires, Saclay, France.     

Pricing problems with a continuum of customers as stochastic Stackelberg games

The pricing problem where a company sells a certain kind of product to a continuum of customers is considered. It is formulated as a stochastic Stackelberg game with nonnested information structure. The inducible region concept, recently developed for deterministic Stackelberg games, is extended to treat the stochastic pricing problem. Necessary and sufficient conditions for a pricing scheme to be optimal are derived, and the pricing problem is solved by first delineating its inducible region, and then solving a constrained optimal control problem.The research work reported here as supported in part by the National Science Foundation under Grant ECS-81-05984, Grant ECS-82-10673, and by the Air Force Office of Scientific Research under AFOSR Grant 80-0098.     

An exponential penalty method for nondifferentiable minimax problems with general constraints

A well-known approach to constrained minimization is via a sequence of unconstrained optimization computations applied to a penalty function. This paper shows how it is possible to generalize Murphy's penalty method for differentiable problems of mathematical programming (Ref. 1) to solve nondifferentiable problems of finding saddle points with constraints. As in mathematical programming, it is shown that the method has the advantages of both Fiacco and McCormick exterior and interior penalty methods (Ref. 2). Under mild assumptions, the method has the desirable property that all trial solutions become feasible after a finite number of iterations. The rate of convergence is also presented. It should be noted that the results presented here have been obtained without making any use of differentiability assumptions.     

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.     

Complex differential games of pursuit-evasion type with state constraints,part 2: Numerical computation of optimal open-loop strategies

In Part 1 of this paper (Ref. 1), necessary conditions for optimal open-loop strategies in differential games of pursuit-evasion type have been developed for problems which involve state variable inequality constraints and nonsmooth data. These necessary conditions lead to multipoint boundary-value problems with jump conditions. These problems can be solved very efficiently and accurately by the well-known multiple-shooting method. By this approach, optimal open-loop strategies and their associated saddle-point trajectories can be computed for the entire capture zone of the game. This also includes the computation of optimal open-loop strategies and saddle-point trajectories on the barrier of the pursuit-evasion game. The open-loop strategies provide an open-loop representation of the optimal feedback strategies. Numerical results are obtained for a special air combat scenario between one medium-range air-to-air missile and one high-performance aircraft in a vertical plane. A dynamic pressure limit for the aircraft imposes a state variable inequality constraint of the first order. Special emphasis is laid on realistic approximations of the lift, drag, and thrust of both vehicles and the atmospheric data. In particular, saddle-point trajectories on the barrier are computed and discussed. Submanifolds of the barrier which separate the initial values of the capture zone from those of the escape zone are computed for two representative launch positions of the missible. By this way, the firing range of the pursuing missile is determined and visualized.This paper is dedicated to the memory of Professor John V. Breakwell.The authors would like to express their sincere and grateful appreciation to Professors R. Bulirsch and K. H. Well for their encouraging interest in this work.     

Complex differential games of pursuit-evasion type with state constraints,part 1: Necessary conditions for optimal open-loop strategies

Complex pursuit-evasion games with state variable inequality constraints are investigated. Necessary conditions of the first and the second order for optimal trajectories are developed, which enable the calculation of optimal open-loop strategies. The necessary conditions on singular surfaces induced by state constraints and non-smooth data are discussed in detail. These conditions lead to multi-point boundary-value problems which can be solved very efficiently and very accurately by the multiple shooting method. A realistically modelled pursuit-evasion problem for one air-to-air missile versus one high performance aircraft in a vertical plane serves as an example. For this pursuit-evasion game, the barrier surface is investigated, which determines the firing range of the missile. The numerical method for solving this problem and extensive numerical results will be presented and discussed in Part 2 of this paper; see Ref. 1.This paper is dedicated to the memory of Professor John V. Breakwell.The authors would like to express their sincere and grateful appreciation to Professors R. Bulirsch and K. H. Well for their encouraging interest in this work.     

Algorithms for uniform optimal strategies in two-player zero-sum stochastic games with perfect information

We deal with zero-sum two-player stochastic games with perfect information. We propose two algorithms to find the uniform optimal strategies and one method to compute the optimality range of discount factors. We prove the convergence in finite time for one algorithm. The uniform optimal strategies are also optimal for the long run average criterion and, in transient games, for the undiscounted criterion as well.     

Optimization of heat flux in domains with temperature constraints

In this paper, we deal with a heat flux optimization problem. We maximize the heat output flow on a portion of a domain's boundary, while on the other portion the distribution of the temperature is fixed. The maximization is carried out under the condition that there are no phase changes.The problem is solved using a convex-functional optimization technique, on Banach spaces, within restricted sets, yielding existence and uniqueness of the solution. The explicit form of the solution and the corresponding Lagrange multipliers associated to the problem are also given.In addition, other optimization problems related to the maximum bound of the heat flux with no phase change are solved.This investigation has been supported by the research and development projects Numerical Analysis of Variational Equalities and Inequalities and Free Boundary Problems in Mathematical Physics from CONICET-UNR, Rosario, Argentina.