Existence and solution methods for equilibria

Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed-point and saddle point problems, and noncooperative games as particular cases. This general format received an increasing interest in the last decade mainly because many theoretical and algorithmic results developed for one of these models can be often extended to the others through the unifying language provided by this common format. This survey paper aims at covering the main results concerning the existence of equilibria and the solution methods for finding them

A class of generalized evolutionary problems driven by variational inequalities and fractional operators

This paper is devoted to a generalized evolution system called fractional partial differential variational inequality which consists of a mixed quasi-variational inequality combined with a fractional partial differential equation in a Banach space. Invoking the pseudomonotonicity of multivalued operators and a generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, first, we prove that the solution set of the mixed quasi-variational inequality involved in system is nonempty, closed and convex. Next, the measurability and upper semicontinuity for the mixed quasi-variational inequality with respect to the time variable and state variable are established. Finally, the existence of mild solutions for the system is delivered. The approach is based on the theory of operator semigroups, the Bohnenblust-Karlin fixed point principle for multivalued mappings, and theory of fractional operators

Differential variational inequalities

International audienceThis paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems , and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensiona

Bounded sets of Lagrange multipliers for vector optimization problems in infinite dimension

AbstractThe aim of this paper is to point out some sufficient constraint qualification conditions ensuring the boundedness of a set of Lagrange multipliers for vectorial optimization problems in infinite dimension. In some (smooth) cases these conditions turn out to be necessary for the existence of multipliers as well