Computing Normalized Equilibria in Convex-Concave Games

Abstract. This paper considers a fairly large class of noncooperative games in which strategies are jointly constrained. When what is called the Ky Fan or Nikaidô-Isoda function is convex-concave, selected Nash equilibria correspond to diagonal saddle points of that function. This feature is exploited to design computational algorithms for finding such equilibria. To comply with some freedom of individual choice the algorithms developed here are fairly decentralized. However, since coupling constraints must be enforced, repeated coordination is needed while underway towards equilibrium. Particular instances include zero-sum, two-person games - or minimax problems - that are convex-concave and involve convex coupling constraints.Noncooperative games; Nash equilibrium; joint constraints; quasivariational inequalities; exact penalty; subgradient projection; proximal point algorithm; partial regularization; saddle points; Ky Fan or Nikaidô-Isoda functions.

Multiobjective Optimization Problems with Equilibrium Constraints

The paper is devoted to new applications of advanced tools of modern variational analysis and generalized differentiation to the study of broad classes of multiobjective optimization problems subject to equilibrium constraints in both finite-dimensional and infinite-dimensional settings. Performance criteria in multiobjectivejvector optimization are defined by general preference relationships satisfying natural requirements, while equilibrium constraints are described by parameterized generalized equations/variational conditions in the sense of Robinson. Such problems are intrinsically nonsmooth and are handled in this paper via appropriate normal/coderivativejsubdifferential constructions that exhibit full calculi. Most of the results obtained are new even in finite dimensions, while the case of infinite-dimensional spaces is significantly more involved requiring in addition certain sequential normal compactness properties of sets and mappings that are preserved under a broad spectrum of operations

Hybrid-type generalized multivalued vector complementarity problems

We introduce a new type of generalized multivalued vector complementarity problems with moving pointed cone. We discuss the existence results for generalized multivalued vector complementarity problems under inclusive assumptions and obtain results on the equivalence between the generalized multivalued vector complementarity problems and the generalized multivalued vector variational inequality problems.Введено новий тип узагальнених багатозначних векторних задач доповнюваностi з рухомим загостреним конусом. Розглянуто питання про iснування розв’язкiв узагальнених багатозначних векторних задач доповнюваностi при умовах включення та отримано результати щодо еквiвалентностi мiж узагальненими багатозначними векторними задачами доповнюваностi та узагальненими багатозначними векторними задачами для варiацiйних нерiвностей

BV-norm continuity of sweeping processes driven by a set with constant shape

We prove the BV-norm well posedness of sweeping processes driven by a moving convex set with constant shape, namely the BV-norm continuity of the so-called play operator of elasto-plasticity.Comment: Minor corrections. To appear in "Journal of Differential Equations

Necessary Conditions in Multiobjective Optimization With Equilibrium Constraints

In this paper we study multiobjective optimization problems with equilibrium constraints (MOECs) described by generalized equations in the form 0 is an element of the set G(x,y) + Q(x,y), where both mappings G and Q are set-valued. Such models particularly arise from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish verifiable necessary conditions for the general problems under consideration and for their important specifications using modern tools of variational analysis and generalized differentiation. The application of the obtained necessary optimality conditions is illustrated by a numerical example from bilevel programming with convex while nondifferentiable data

Optimization and Equilibrium Problems with Equilibrium Constraints in Infinite-Dimensional Spaces

The paper is devoted to applications of modern variational f).nalysis to the study of constrained optimization and equilibrium problems in infinite-dimensional spaces. We pay a particular attention to the remarkable classes of optimization and equilibrium problems identified as MPECs (mathematical programs with equilibrium constraints) and EPECs (equilibrium problems with equilibrium constraints) treated from the viewpoint of multiobjective optimization. Their underlying feature is that the major constraints are governed by parametric generalized equations/variational conditions in the sense of Robinson. Such problems are intrinsically nonsmooth and can be handled by using an appropriate machinery of generalized differentiation exhibiting a rich/full calculus. The case of infinite-dimensional spaces is significantly more involved in comparison with finite dimensions, requiring in addition a certain sufficient amount of compactness and an efficient calculus of the corresponding sequential normal compactness (SNC) properties

Variational inequalities and optimization problems

In this paper we survey the relationships between scalar and vector variational inequalities (of differential type) and the underlying optimizationproblem. We show that the variational inequalities of Stampacchia type can be viewed as necessary optimality conditions, while the variational inequalities of Minty type can be considered as sufficient optimality conditions. Concerning this last statement, a gap is observed between the scalar and the vector case and possible fulfilments of this gap are investigated

Advances in Optimization and Nonlinear Analysis

The present book focuses on that part of calculus of variations, optimization, nonlinear analysis and related applications which combines tools and methods from partial differential equations with geometrical techniques. More precisely, this work is devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The book is a valuable guide for researchers, engineers and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science and economics