Van Hien Nguyen

Extragradient algorithms extended to equilibrium problems

We make use of the auxiliary problem principle to develop iterative algorithms for solving equilibrium problems. The first one is an extension of the extragradient algorithm to equilibrium problems. In this algorithm the equilibrium bifunction is not required to satisfy any monotonicity property, but it must satisfy a certain Lipschitz-type condition. To avoid this requirement we propose linesearch procedures commonly used in variational inequalities to obtain projection-type algorithms for solving equilibrium problems. Applications to mixed variational inequalities are discussed. A special class of equilibrium problems is investigated and some preliminary computational results are reported. This article is dedicated to the Memory of W. Oettli.

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.

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.

A Generalized Method of Moving Asymptotes (GMMA) Including Equality Constraints

A convex programming optimizer called GMMA (Generalized Method of Moving Asymptotes) is presented in this paper. This method aims at solving engineering design problems including nonlinear equality and inequality constraints. The basic feature of this optimizer is that the efficient dual solution strategy together with the flexible GMMA approximation scheme are used. Especially, nonlinear equality constraints can be exactly satisfied by the intermediate solution of each explicit subproblem because their linearization is updated in an internal loop of the subproblem. This method will be illustrated by a hydrodynamic design application.

On Nash---Cournot oligopolistic market equilibrium models with concave cost functions

We consider Nash–Cournot oligopolistic market equilibrium models with concave cost functions. Concavity implies, in general, that a local equilibrium point is not necessarily a global one. We give conditions for existence of global equilibrium points. We then propose an algorithm for finding a global equilibrium point or for detecting that the problem is unsolvable. Numerical experiments on some randomly generated data show efficiency of the proposed algorithm.

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.