Michel Théra

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:

Metric Inequality, Subdifferential Calculus and Applications

(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,

Error Bounds and Implicit Multifunction Theorem in Smooth Banach Spaces and Applications to Optimization

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.

Error bounds for systems of lower semicontinuous functions in Asplund spaces

In this note we present a new result which is equivalent to the celebrated Ekelands variational principle, and a set of implications which includes a new non-convex minimisation principle due to Takahashi.

Error Bounds in Metric Spaces and Application to the Perturbation Stability of Metric Regularity

In this paper we establish characterizations of Asplund spaces in terms of conditions ensuring the metric inequality and intersection formulae. Then we establish chain rules for the limiting Fréchet subdifferentials. Necessary conditions for constrained optimization problems with non-Lipschitz data are derived.

Conjugate convex operators

In this paper, the equivalence between variational inclusions and a generalized type of Weiner–Hopf equation is established. This equivalence is then used to suggest and analyze iterative methods in order to find a zero of the sum of two maximal monotone operators. Special attention is given to the case where one of the operators is Lipschitz continuous and either is strongly monotone or satisfies the Dunn property. Moreover, when the problem has a nonempty solution set, a fixed-point procedure is proposed and its convergence is established provided that the Brézis–Crandall–Pazy condition holds true. More precisely, it is shown that this allows reaching the element of minimal norm of the solution set.

Stability of Error Bounds for Convex Constraint Systems in Banach Spaces

In this paper we provide an error bound estimate and an implicit multifunction theorem in terms of smooth subdifferentials and abstract subdifferentials. Then, we derive a subdifferential calculus and Fritz–John type necessary optimality conditions for constrained minimization problems.

Enlargements and sums of monotone operators

In this paper, using the Fréchet subdifferential, we derive several sufficient conditions ensuring an error bound for inequality systems in Asplund spaces. As an application we obtain in the context of Banach spaces a global error bound for quadratic nonconvex inequalities and we derive necessary optimality conditions for optimization problems.