Global saddle-point duality for quasi-concave programs,II

A saddle-point duality is proposed for the quasi-concave non-differentiable case of the maximization of the minimum between a finite number of functions. This class of problems contains quasi-concave (convex) programs that are known to be irreducible to convex ones. With the aid of the saddle-point duality involving conjugate-like operators, a Lagrangian is presented, the saddle-points of which give the exact global solutions. A few particular cases are discussed, among them the Von Neumann economic model and discrete rational approximation.     

Singular linear quadratic differential games with bounds on control

A class of two-persons zero-sum differential games with the possibility of singular arcs is considered. Under some assumptions, a sufficient condition for the existence of a saddle-point solution is derived. In this approach, the properties of convexity and concavity of the performance index are used to find the sufficient condition. It is observed that, unlike total singularity, the occurrence of partially singular arcs does not cause any problems for the existence of a unique saddle-point solution.     

Zero-sum risk-sensitive stochastic games on a countable state space

Infinite horizon discounted-cost and ergodic-cost risk-sensitive zero-sum stochastic games for controlled Markov chains with countably many states are analyzed. Upper and lower values for these games are established. The existence of value and saddle-point equilibria in the class of Markov strategies is proved for the discounted-cost game. The existence of value and saddle-point equilibria in the class of stationary strategies is proved under the uniform ergodicity condition for the ergodic-cost game. The value of the ergodic-cost game happens to be the product of the inverse of the risk-sensitivity factor and the logarithm of the common Perron–Frobenius eigenvalue of the associated controlled nonlinear kernels.     

Some thoughts on saddle-point conditions and information structures in zero-sum differential games

For a very simple two-stage, linear-quadratic, zero-sum difference game with dynamic information structure, we show that (i) there exist nonlinear saddle-point strategies which require the same existence conditions as the well-known linear, closed-loop, no-memory solution and (ii) there exist both linear and nonlinear saddle-point strategies which require more stringent conditions than the unique open-loop solution. We then discuss the implication of this result with respect to the existence of saddle points in zero-sum differential games for different information patterns.     

Existence of optimal open-loop strategies for a class of differential games

In this paper, the class of differential games with linear system equations and a quadratic performance index is investigated for saddlepoint solutions when one or both of the players use open-loop control. For each formulation of the game, a necessary and sufficient condition is obtained for the existence of an optimal strategy pair that generates a regular optimal path. For those cases where a solution exists, the unique saddle-point solution is presented. Also, relationships are established between the time intervals of existence of solutions for the various formulations of the game.This research was supported by the National Science Foundation, Grant No. GK-3341.     

Saddle-point solutions to Dirichlet problems on the Sierpiński gasket

We provide sufficient conditions for the existence of saddle-point solutions to a system driven by the weak Laplacian on the Sierpiński gasket. We analyze also its stability by proving its continuous dependence on parameters.     

Zero-sum games for continuous-time Markov jump processes with risk-sensitive finite-horizon cost criterion

In this paper we study the zero-sum games for continuous-time Markov jump processes under the risk-sensitive finite-horizon cost criterion. The state space is a Borel space and the transition rates are allowed to be unbounded. Under the suitable conditions, we use a new value iteration approach to establish the existence of a solution to the risk-sensitive finite-horizon optimality equations of the players, obtain the existence of the value of the game and show the existence of saddle-point equilibria.     

Saddle-point equilibrium sequence in one class of singular infinite horizon zero-sum linear-quadratic differential games with state delays

ABSTRACT

We consider an infinite horizon zero-sum linear-quadratic differential game with state delays in the dynamics. The cost functional of this game does not contain a control cost of the minimizing player (the minimizer), meaning that the considered game is singular. For this game, definitions of the saddle-point equilibrium and the game value are proposed. These saddle-point equilibrium and game value are obtained by a regularization of the singular game. Namely, we associate this game with a new differential game for the same equation of dynamics. The cost functional in the new game is the sum of the original cost functional and an infinite horizon integral of the square of the minimizer's control with a small positive weight coefficient. This new game is regular, and it is a cheap control game. An asymptotic analysis of this cheap control game is carried out. Using this asymptotic analysis, the existence of the saddle-point equilibrium and the value of the original game is established, and their expressions are derived. Illustrative example is presented.     

Constrained markov decision processes with compact state and action spaces: the average case

Constrained Markov decision processes with compact state and action spaces are studied under long-run average reward or cost criteria. By introducing a corresponding Lagrange function, a saddle-point theorem is given, by which the existence of a constrained optimal pair of initial state distribution and policy is shown. Also, under the hypothesis of Doeblin, the functional characterization of a constrained optimal policy is obtained     

Zero-sum risk-sensitive stochastic games for continuous time Markov chains

We study infinite horizon discounted-cost and ergodic-cost risk-sensitive zero-sum stochastic games for controlled continuous time Markov chains on a countable state space. For the discounted-cost game, we prove the existence of value and saddle-point equilibrium in the class of Markov strategies under nominal conditions. For the ergodic-cost game, we prove the existence of values and saddle point equilibrium by studying the corresponding Hamilton-Jacobi-Isaacs equation under a certain Lyapunov condition.     

Substructuring preconditioners for saddle-point problems arising from Maxwell's equations in three dimensions

This paper is concerned with the saddle-point problems arising from edge element discretizations of Maxwell's equations in a general three dimensional nonconvex polyhedral domain. A new augmented technique is first introduced to transform the problems into equivalent augmented saddle-point systems so that they can be solved by some existing preconditioned iterative methods. Then some substructuring preconditioners are proposed, with very simple coarse solvers, for the augmented saddle-point systems. With the preconditioners, the condition numbers of the preconditioned systems are nearly optimal; namely, they grow only as the logarithm of the ratio between the subdomain diameter and the finite element mesh size.

     

General KKM Theorem with Applications to Minimax and Variational Inequalities

In this paper, a general version of the KKM theorem is derived by using the concept of generalized KKM mappings introduced by Chang and Zhang. By employing our general KKM theorem, we obtain a general minimax inequality which contains several existing ones as special cases. As applications of our general minimax inequality, we derive an existence result for saddle-point problems under general setting. We also establish several existence results for generalized variation inequalities and generalized quasi-variational inequalities.     

Optimal estimation and control of discrete multiplicative systems with unknown second-order statistics

The problem of estimation and control for discrete-time systems with multiplicative noise is examined. Such systems occur naturally in the modeling of stochastic systems with random or unknown coefficients and appear to be robust in contrast to LQG regulators which are sensitive to errors in the coefficients.The statistics of the white sequences of the system are unknown. The problem of stochastic estimation and control of such a system is difficult not only because of the unknown statistics but also because the state is not Gaussian.The approach of this work is to convert the stochastic problem to a deterministic game-theoretic one. We find the estimator and controller so as to minimize a suitable performance measure assuming the worst behavior of nature.A set of necessary and sufficient conditions is developed for the existence of a saddle-point estimator. When both estimation and control are considered, two difficulties appear: the optimality conditions are only necessary and the separation principle collapses. As a result, the saddle-point conditions are only necessary. If the covariances belong to sets with maximal points, then the necessary conditions are satisfied at these points. If, on the other hand, they belong to convex and compact sets and the system has a steady state, then the estimation problem alone has always a saddle-point solution.     

Nonlinear Separation Approach to Constrained Extremum Problems

In this paper, by virtue of a nonlinear scalarization function, two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function are first introduced, respectively. Then, by the image space analysis, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, some necessary and sufficient optimality conditions are obtained for constrained extremum problems.     

Minimax principles for elliptic mixed hemivariational–variational inequalities

In this paper, minimax principles are explored for elliptic mixed hemivariational–variational inequalities. Under certain conditions, a saddle-point formulation is shown to be equivalent to a mixed hemivariational–variational inequality. While the minimax principle is of independent interest, it is employed in this paper to provide an elementary proof of the solution existence of the mixed hemivariational–variational inequality. Theoretical results are illustrated in the applications of two contact problems.     

Risk-sensitive stochastic differential games with reflecting diffusions

We study risk-sensitive differential games for controlled reflecting diffusion processes in a bounded domain. We consider both nonzero-sum and zero-sum cases. We treat two cost evaluation criteria; namely, discounted cost and ergodic cost. Under certain assumptions we establish the existence of Nash/saddle-point equilibria for relevant cases.     

Globally proper saddle point in ic-cone-convexlike set-valued optimization problems

The existence conditions of globally proper efficient points and a useful property of ic- cone-convexlike set-valued maps are obtained. Under the assumption of the ic-cone-convexlikeness, the optimality conditions for globally proper efficient solutions are established in terms of Lagrange multipliers. The new concept of globally proper saddle-point for an appropriate set-valued Lagrange map is introduced and used to characterize the globally proper efficient solutions. The results which are obtained in this paper are proven under the conditions that the ordering cone need not to have a nonempty interior.     

Existence of Value and Saddle Point in Infinite-Dimensional Differential Games

We study two-player zero-sum differential games of finite duration in a Hilbert space. Following the Berkovitz notion of strategies, we prove the existence of value and saddle-point equilibrium. We characterize the value as the unique viscosity solution of the associated Hamilton–Jacobi–Isaacs equation using dynamic programming inequalities.     

风险的动态定量和一个相关的随机对策问题

本文讨论不完全市场中股票收益率不确定时的动态风险度量问题和一个相关的随机对策问题。该动态风险度量可表示为一个随机最优控制问题的值函数,以倒向随机微分方程为工具我们给出了最优目标具有的形式,并给出随机对策问题上值与下值相等的充分条件和鞍点的存在性。     

On the saddle-point solution of a class of stochastic differential games

This paper deals with the saddle-point solution of a class of stochastic differential games described by linear state dynamics and quadratic objective functionals. The information structure of the problem is such that both players have access to a common noisy linear measurement of the state and they are permitted to utilize only this information in constructing their controls. The saddle-point solution of such differential game problems has been discussed earlier in Ref. 1, but the conclusions arrived there are incorrect, as is explicitly shown in this paper. We extensively discuss the role of information structure on the saddle-point solution of such stochastic games (specifically within the context of an illustrative discrete-time example) and then obtain the saddle-point solution of the problem originally formulated by employing an indirect approach.This work was done while the author was on sabbatical leave at Twente University of Technology, Department of Applied Mathematics, Enschede, Holland, from Applied Mathematics Division, Marmara Scientific and Industrial Research Institute, Gebze, Kocaeli, Turkey.