Jean-Jacques Strodiot

Convergence analysis and applications of the Glowinski-Le Tallec splitting method for finding a zero of the sum of two maximal monotone operators

Many problems of convex programming can be reduced to finding a zero of the sum of two maximal monotone operators. For solving this problem, there exists a variety of methods such as the forward–backward method, the Peaceman–Rachford method, the Douglas–Rachford method, and more recently the θ-scheme. This last method has been presented without general convergence analysis by Glowinski and Le Tallec and seems to give good numerical results. The purpose of this paper is first to present convergence results and an estimation of the rate of convergence for this recent method, and then to apply it to variational inequalities and structured convex programming problems to get new parallel decomposition algorithms.

A Bundle Method for Solving Variational Inequalities

In this paper, we present a bundle method for solving a generalized variational inequality problem. This problem consists of finding a zero of the sum of two multivalued operators defined on a real Hilbert space. The first one, F, is monotone and the second is the subdifferential of a lower semicontinuous proper convex function. Our method is based on the auxiliary problem principle due to Cohen, and our strategy is to approximate, in the subproblems, the nonsmooth convex function by a sequence of convex piecewise linear functions, as in the bundle method for nonsmooth optimization. This makes the subproblems more tractable. First, we explain how to build, step by step, suitable piecewise linear approximations by means of a bundle strategy, and we present a new stopping criterion to determine whether the current approximation is good enough. This criterion is the same as that commonly used in the special case of nonsmooth optimization. Second, we study the convergence of the algorithm for the case when the stepsizes are chosen going to zero and for the case bounded away from zero. In the first case, the convergence can be proved under rather mild assumptions: the operator F is paramonotone and possibly multivalued. In the second case, the convergence needs a stronger assumption: F is single-valued and satisfies a Dunn property. Finally, we illustrate the behavior of the proposed algorithm by some numerical tests.

A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems

Generalized Nash equilibrium problems are important examples of quasi-equilibrium problems. The aim of this paper is to study a general class of algorithms for solving such problems. The method is a hybrid extragradient method whose second step consists in finding a descent direction for the distance function to the solution set. This is done thanks to a linesearch. Two descent directions are studied and for each one several steplengths are proposed to obtain the next iterate. A general convergence theorem applicable to each algorithm of the class is presented. It is obtained under weak assumptions: the pseudomonotonicity of the equilibrium function and the continuity of the multivalued mapping defining the constraint set of the quasi-equilibrium problem. Finally some preliminary numerical results are displayed to show the behavior of each algorithm of the class on generalized Nash equilibrium problems.

The interior proximal extragradient method for solving equilibrium problems

In this article we present a new and efficient method for solving equilibrium problems on polyhedra. The method is based on an interior-quadratic proximal term which replaces the usual quadratic proximal term. This leads to an interior proximal type algorithm. Each iteration consists in a prediction step followed by a correction step as in the extragradient method. In a first algorithm each of these steps is obtained by solving an unconstrained minimization problem, while in a second algorithm the correction step is replaced by an Armijo-backtracking linesearch followed by an hyperplane projection step. We prove that our algorithms are convergent under mild assumptions: pseudomonotonicity for the two algorithms and a Lipschitz property for the first one. Finally we present some numerical experiments to illustrate the behavior of the proposed algorithms.

Computing a global optimal solution to a design centering problem

In this paper we present a method for solving a special three-dimensional design centering problem arising in diamond manufacturing: Find inside a given (not necessarily convex) polyhedral rough stone the largest diamond of prescribed shape and orientation. This problem can be formulated as the one of finding a global maximum of a difference of two convex functions over ℝ3 and can be solved efficiently by using a global optimization algorithm provided that the objective function of the maximization problem can be easily evaluated. Here we prove that with the information available on the rough stone and on the reference diamond, evaluating the objective function at a pointx amounts to computing the distance, with respect to a Minkowski gauge, fromx to a finite number of planes. We propose a method for finding these planes and we report some numerical results.

Coupling the auxiliary problem principle and epiconvergence theory to solve general variational inequalities

Many algorithms for solving variational inequality problems can be derived from the auxiliary problem principle introduced several years ago by Cohen. In recent years, the convergence of these algorithms has been established under weaker and weaker monotonicity assumptions: strong (pseudo) monotonicity has been replaced by the (pseudo) Dunn property. Moreover, well-suited assumptions have given rise to local versions of these results.In this paper, we combine the auxiliary problem principle with epiconvergence theory to present and study a basic family of perturbed methods for solving general variational inequalities. For example, this framework allows us to consider barrier functions and interior approximations of feasible domains. Our aim is to emphasize the global or local assumptions to be satisfied by the perturbed functions in order to derive convergence results similar to those without perturbations. In particular, we generalize previous results obtained by Makler-Scheimberg et al.

A MATHEMATICAL CONVERGENCE ANALYSIS OF THE CONVEX LINEARIZATION METHOD FOR ENGINEERING DESIGN OPTIMIZATION

Abstract This paper is concerned with the convex linearization method recently proposed by Fleury and Braibant for structural optimization. We give here a mathematical convergence analysis or this method. We also discuss some modifications of it.

On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space

Abstract In this paper, new numerical algorithms are introduced for finding the solution of a variational inequality problem whose constraint set is the common elements of the set of fixed points of a demicontractive mapping and the set of solutions of an equilibrium problem for a monotone mapping in a real Hilbert space. The strong convergence of the iterates generated by these algorithms is obtained by combining a viscosity approximation method with an extragradient method. First, this is done when the basic iteration comes directly from the extragradient method, under a Lipschitz-type condition on the equilibrium function. Then, it is shown that this rather strong condition can be omitted when an Armijo-backtracking linesearch is incorporated into the extragradient iteration. The particular case of variational inequality problems is also examined.

On the Contraction and Nonexpansiveness Properties of the Marginal Mappings in Generalized Variational Inequalities Involving Co-Coercive Operators

We investigate the contraction and nonexpansiveness properties of the marginal mappings for gap functions in generalized variational inequalities dealing with strongly monotone and co-coercive operators in a real Hilbert space. We show that one can choose regularization operators such that the solution of a strongly monotone variational inequality can be obtained as the fixed point of a certain contractive mapping. Moreover a solution of a co-coercive variational inequality can be computed by finding a fixed point of a certain nonexpansive mapping. The results give a further analysis for some methods based on the auxiliary problem principle. They also lead to new algorithms for solving generalized variational inequalities involving co-coercive operators. By the Banach contraction mapping principle the convergence rate can be easily established.

A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces

A new class of extragradient-type methods is introduced for solving an equilibrium problem in a real Hilbert space without any monotonicity assumption on the equilibrium function. The strategy is to replace the second projection step in the classical extragradient method by a projection onto shrinking convex subsets of the feasible set. Furthermore, to ensure a sufficient decrease on the equilibrium function, a general Armijo-type condition is imposed. This condition is shown to be satisfied for four different linesearches used in the literature. Then, the weak and strong convergence of the resulting algorithms is obtained under non-monotonicity assumptions. Finally, some numerical experiments are reported.