Abdellatif Moudafi

Proximal and dynamical approaches to equilibrium problems

The theory of equilibrium problems has emerged as an interesting branch of applied mathematics, permitting the general and unified study of a large number of problems arising in mathematical economics, optimization and operations research. Inspired by numerical methods developed for variational inequalities and motivated by recent advances in this field, we propose several ways (including an auxiliary problem principle, a selection method, as well as a dynamical procedure) to solve the following equilibrium problem:

Convergence of a splitting inertial proximal method for monotone operators

(GEP)Find\overline x \in CsuchthatF(\overline x ,x) + \left\langle {G(\overline x ),x - \overline x } \right\rangle \geqslant 0\forall x \in C,

Proximal methods for a class of bilevel monotone equilibrium problems

where C is a nonempty convex closed subset of a real Hilbert space X, F: C × C → ℝ is a given bivariate function with F(x, x) = 0 for all x ∈ C and G: C → ℝ is a continuous mapping. This problem has useful applications in nonlinear analysis, including as special cases optimization problems, variation al inequalities, fixed-point problems and problems of Nash equilibria. Throughtout the paper, X is a real Hilbert space, denotes the associated inner product and | · | stands for the corresponding norm. From now on, we assume that the solution set, S, of problem (GEP) is nonempty. This corresponds to some important situations such as linear programming and semi-coercive minimization problems.

Combining The Proximal Algorithm And Tikhonov Regularization

Based on the very recent work by Censor-Gibali-Reich (http://arxiv.org/abs/1009.3780), we propose an extension of their new variational problem (Split Variational Inequality Problem) to monotone variational inclusions. Relying on the Krasnosel’skii-Mann Theorem for averaged operators, we analyze an algorithm for solving new split monotone inclusions under weaker conditions. Our weak convergence results improve and develop previously discussed Split Variational Inequality Problems, feasibility problems and related problems and algorithms.

Finding a zero of the sum of two maximal monotone operators

A forward-backward inertial procedure for solving the problem of finding a zero of the sum of two maximal monotone operators is proposed and its convergence is established under a cocoercivity condition with respect to the solution set.

Towards viscosity approximations of hierarchical fixed-point problems.

Abstract The aim of this paper is twofold. First, it is to extend the sensitivity analysis framework, developed recently for variational inequalities, to mixed equilibrium problems. The second is to propose iterative methods for solving this kind of problems. In the process, we establish an equivalence between an extended version of Wiener-Hopf equations and the given problems relying on a generalization of the Yosida approximation notion. Our results generalize results obtained for optimization, variational inequalities, complementarity problems, and problems of Nash equilibria.

Convergence of New Inertial Proximal Methods for DC Programming

We consider a bilevel problem involving two monotone equilibrium bifunctions and we show that this problem can be solved by a simple proximal method. Under mild conditions, the weak convergence of the sequences generated by the algorithm is obtained. Using this result we obtain corollaries which improve several corresponding results in this field.

SENSITIVITY ANALYSIS FOR VARIATIONAL INCLUSIONS BY WIENER-HOPF EQUATION TECHNIQUES

An approximatio n meth od which com bines Tikh on ov method with the proximal point a lgo rithm, is presen ted. Co nditions which gua ra ntee the convergence to a pa rt icula r element of the so lutio n set, ar e pro vided. A particular atte ntion is given to the co nvex a nd convex-co ncave optimiza tion cases