Sensitivity analysis for generalized equations

We focus on the study of sensitivity analysis for generalized equations, permitting to obtain important information on the behavior of the solution mapping of parametric variational inequalities and parametric equilibrium problems. The topic belongs to the wider field of nonlinear variational analysis with many practical applications. The project proposal targets three objectives, briefly described in what follows. 1. Linear openness and metric regularity: applications to parametric problems. Three equivalent properties, linear openness, metric regularity and Aubin property give information on the behavior of single-valued or set-valued mappings in a neighbourhood of a point in their graph. In particular, linear openness implies surjectivity, which assures the existence of solutions to generalized equations. Moreover, Aubin property provides information about the behavior of the solution sets of these equations. Our goal is to obtain new results concerning the openness of the sum, or more general, the composition of set-valued mappings, which allow us to deduce calmness and Lipschitz-likeness of the solution maps for parametric variational inequalities and parametric equilibrium problems. The way to obtain them is to transform the latter problems into generalized equations by means of the generating operator, and the diagonal subdifferential operator, respectively. 2. Ekeland type variational principles: a new approach for equilibrium problems. This principle is an important tool in proving openness and metric regularity. In the last decades many results have been published concerning different extensions of this principle for different kinds of problems, including the equilibrium problem. The basic assumption which occurs within these results is the triangle inequality for the equilibrium bifunction, a rather demanding condition that rarely holds (take for instance the case of variational inequalities). Therefore, one of our goals is to relax the triangle inequality by means of the so-called cyclic antimonotonicity for vector bifunctions. Then, we intend to establish an approximate vector Ekelands variational principle when the domain of the bifunction is a (not necessarily complete) metric space, and its range is a vector space without any particular topology. The expected results will provide a unified approach to various existing results in the literature; among them, those of Ansari et al. [AHY] and Lin et al. [LCH]. 3. Numerical methods for variational inequalities, saddle-point and equilibrium problems. Based on the so called extragradient algorithms introduced by Korpelevich [KOR], we propose to construct new efficient methods for solving variational inequalities, saddle-point, or equilibrium problems, by showing their weak and (under some extra assumptions) strong convergence in reflexive Banach spaces. We intend to extend some existing algorithms from Hilbert spaces to a reflexive Banach space setting by making use of Bregman functions and Bregman distances. It is also our goal to compare the new results with the proximal point algorithm for equilibrium problems developed by Burachik and Kassay [BK]), and with the algorithms established by Ansari et al. [ACWY, CAPY1, CAPY2]). At this point, a challenging issue would be to implement these algorithms in order to obtain numerical results for problems of practical interest.