On Nash Equilibria for Noncooperative Games Governed by the Burgers Equation

Existence of a Nash equilibrium in a noncooperative game governed by the one-dimensional Burgers equation, proposed in the case of pointwise controls in Ref. 1, is proved under data qualifications that guarantee the diffusion term in the Burgers’ equation to be dominant enough with respect to the uniform convexity of the payoffs. This work was partly supported by Grants 201/03/0934 (GA čR) and MSM 0021620839 (MšMT čR). Inspiring discussions with Angel M. Ramos are acknowledged.     

A class of nondifferentiable control problems

Optimality conditions and duality results are obtained for a class of control problems having a nondifferentiable term in the integrand of the objective functional. These results generalize many well-known results in optimal control theory involving differentiable functions, and also provide a relationship with certain nondifferentiable mathematical programming problems. Some extensions concerning the unified treatment of optimal control theory and continuous programming are also mentioned. Finally, a control problem containing an arbitrary norm, along with its appropriate norm, is given.     

A note on optimality conditions for the Euclidean. Multifacility location problem

Thekey problem of the Euclidean multifacility location (EMFL) problem is to decide whether a givendead point is optimal. If it is not optimal, we wish to compute a descent direction. This paper extends the optimality conditions of Calamai and Conn and Overton to the case when the rows of the active constraints matrix are linearly dependent. We show that linear dependence occurs wheneverG, the graph of the coinciding facilities, has a cycle. In this case the key problem is formulated as a linear least squares problem with bounds on the Euclidean norms of certain subvectors.     

First and Second Order Optimality Conditions Using Approximations for Nonsmooth Vector Optimization in Banach Spaces

We use the first and second order approximations of mappings to establish both necessary and sufficient optimality conditions for unconstrained and constrained nonsmooth vector optimization problems. Ideal solutions, efficient solutions, and weakly efficient solutions are considered. The data of the problems need not even be continuous. Some often imposed compactness assumptions are also relaxed. Examples are provided to compare our results and some known recent results.This work was partially supported by the National Basic Research Program in Natural Sciences of Vietnam.     

Smooth and Nonsmooth Lipschitz Controls for a Class of Vector Differential Equations

This paper is devoted to the study of a class of control problems associated to a nonlinear second-order vector differential equation with pointwise state constraints. The control is realized via a function of the state. We extend the results of Akkouchi, Bounabat, and Goebel to vector differential equations and furthermore consider the more general case. Under proper conditions, we prove the existence of optimal controls in the class of Lipschitz functions and obtain an optimality condition which looks somehow like the Pontryagin maximum principle for a smooth optimal control function. For a nonsmooth optimal control function, we derive a suboptimality condition by means of the Ekeland variational principle.Communicated by M. J. BalasThis work was supported by 985 Project of Jilin University. The author thanks Professor Yong Li for valuable suggestions. He also thanks Professor M. J. Balas and the anonymous referees for their comments.     

CONE-DIRECTED CONTINGENT DERIVATIVES AND GENERALIZED PREINVEX SET-VALUED OPTIMIZATION

By using cone-directed contingent derivatives, the unified necessary and suffi-cient optimality conditions are given for weakly and strongly minimal elements respectively in generalized preinvex set valued optimization.     

Contingent epiderivatives and set-valued optimization

In this paper we introduce the concept of the contingent epiderivative for a set-valued map which modifies a notion introduced by Aubin [2] as upper contingent derivative. It is shown that this kind of a derivative has important properties and is one possible generalization of directional derivatives in the single-valued convex case. For optimization problems with a set-valued objective function optimality conditions based on the concept of the contingent epiderivative are proved which are necessary and sufficient under suitable assumptions.     

Directional derivatives in nonsmooth optimization

In this note, we consider two notions of second-order directional derivatives and discuss their use in the characterization of minimal points for nonsmooth functions.This research was supported by NSF Grant No. ECS-8214081, by the Fund for Promotion of Research at the Technion, and by Deutsche Forschungsgemeinschaft.     

Second-Order Optimality Conditions in Set Optimization

In this paper, we propose second-order epiderivatives for set-valued maps. By using these concepts, second-order necessary optimality conditions and a sufficient optimality condition are given in set optimization. These conditions extend some known results in optimization.The authors are grateful to the referees for careful reading and helpful remarks.     

The contribution of K.-H. Elster to generalized conjugation theory and nonconvex duality

This article surveys the main contributions of K.-H. Elster to the theory of generalized conjugate functions and its applications to duality in nonconvex optimization.