Gap Functions For Variational Inequalities Research Articles

Weak sharp solutions for generalized variational inequalities

We consider weak sharp solutions for the generalized variational inequality problem, in which the underlying mapping is set-valued, and not necessarily monotone. We extend the concept of weak sharpness to this more general framework, and establish some of its characterizations. We establish connections between weak sharpness and (1) gap functions for variational inequalities, and (2) global error bound. When the solution set is weak sharp, we prove finite convergence of the sequence generated by an arbitrary algorithm, for the monotone set-valued case, as well as for the case in which the underlying set-valued map is either Lipschitz continuous in the set-valued sense, for infinite dimensional spaces, or inner-semicontinuous when the space is finite dimensional.

Revisiting the construction of gap functions for variational inequalities and equilibrium problems via conjugate duality

Abstract Based on conjugate duality we construct several gap functions for general variational inequalities and equilibrium problems, in the formulation of which a so-called perturbation function is used. These functions are written with the help of the Fenchel-Moreau conjugate of the functions involved. In case we are working in the convex setting and a regularity condition is fulfilled, these functions become gap functions. The techniques used are the ones considered in [Altangerel L., Boţ R.I., Wanka G., On gap functions for equilibrium problems via Fenchel duality, Pac. J. Optim., 2006, 2(3), 667–678] and [Altangerel L., Boţ R.I., Wanka G., On the construction of gap functions for variational inequalities via conjugate duality, Asia-Pac. J. Oper. Res., 2007, 24(3), 353–371]. By particularizing the perturbation function we rediscover several gap functions from the literature. We also characterize the solutions of various variational inequalities and equilibrium problems by means of the properties of the convex subdifferential. In case no regularity condition is fulfilled, we deliver also necessary and sufficient sequential characterizations for these solutions. Several examples are illustrating the theoretical aspects.

Regularized gap function as penalty term for constrained minimization problems

By using the regularized gap function for variational inequalities, Li and Peng introduced a new penalty function P α ( x ) for the problem of minimizing a twice continuously differentiable function in closed convex subset of the n-dimensional space R n . Under certain assumptions, they proved that the original constrained minimization problem is equivalent to unconstrained minimization of P α ( x ) . The main purpose of this paper is to give an in-depth study of those properties of the objective function that can be extended from the feasible set to the whole R n by P α ( x ) . For example, it is proved that the objective function has bounded level sets (or is strongly coercive) on the feasible set if and only if P α ( x ) has bounded level sets (or is strongly coercive) on R n . However, the convexity of the objective function does not imply the convexity of P α ( x ) when the objective function is not quadratic, no matter how small α is. Instead, the convexity of the objective function on the feasible set only implies the invexity of P α ( x ) on R n . Moreover, a characterization for the invexity of P α ( x ) is also given.

ON THE CONSTRUCTION OF GAP FUNCTIONS FOR VARIATIONAL INEQUALITIES VIA CONJUGATE DUALITY

In this paper, we deal with the construction of gap functions for variational inequalities by using an approach which bases on the conjugate duality. Under certain assumptions we also investigate a further class of gap functions for the variational inequality problem, the so-called dual gap functions.

Exact Penalty Functions for Constrained Minimization Problems via Regularized Gap Function for Variational Inequalities

By using the regularized gap function for variational inequalities, we introduce a new penalty function P ?(x) for the problem of minimizing a twice continuously differentiable function in a closed convex subset of the n-dimensional space $$\mathbb{R}^n$$ . Under certain assumptions, it is shown that any stationary point of the penalty function P ?(x) satisfies the first-order optimality condition of the original constrained minimization problem, and any local (or global) minimizer of P ?(x) on $$\mathbb{R}^n$$ is a locally (or globally) optimal solution of the original optimization problem.

On finite-dimensional generalized variational inequalities

Our aim is to provide a short analysis of the generalized variational inequality (GVI) problem from both theoretical and algorithmic points of view. First, we show connections among some well known existence theorems for GVI and for inclusions. Then, we recall the proximal point approach and a splitting algorithm for solving GVI. Finally, we propose a class of differentiable gap functions for GVI, which is a natural extension of a well known class of gap functions for variational inequalities (VI).

Generalized Differential Properties of the Auslender Gap Function for Variational Inequalities

In this note, the Auslender gap function, which is used to formulate a variational inequality into an equivalent minimization problem, is shown to be differentiable in the generalized sense and has a lower contingent derivative under suitable conditions. This enables us to establish necessary and sufficient conditions for the existence of a solution to problems of variational inequalities.

The Dual Gap Function for Variational Inequalities

In this paper we further study the dual gap function G, which was introduced by Marcotte and Zhu, for the variational inequality problem (VIP). We characterize the directional derivative and subdifferential of G. Based on these, we get a better understanding of the concepts of a global error bound, weak sharpness, and minimum principle sufficiency property for the pseudo-monotone.

New existence results for equilibrium problems

We consider equilibrium problems in the framework of the formulation proposed by Blum and Oettli, which includes variational inequalities, Nash equilibria in noncooperative games, and vector optimization problems, for instance, as particular cases. We establish new sufficient and/or necessary conditions for existence of solutions of such problems. Our results are based upon the relation between equilibrium problems and certain auxiliary convex feasibility problems, together with extensions to equilibrium problems of gap functions for variational inequalities. Then we apply our results to some particular instances of equilibrium problems, obtaining results which include, among others, a new lemma of the alternative for convex optimization problems.

On convergence of descent methods for variational inequalities in a Hilbert space

In this paper, properties of differentiable gap functions for variational inequalities and convergence of the corresponding descent methods under a Hilbert space setting are considered. We give various convergence results under different assumptions on the cost mapping, including the monotone case.

Weak Sharp Solutions of Variational Inequalities

Weak Sharp Solutions of Variational Inequalities

A class of gap functions for variational inequalities

Recently Auchmuty (1989) has introduced a new class of merit functions, or optimization formulations, for variational inequalities in finite-dimensional space. We develop and generalize Auchmuty's results, and relate his class of merit functions to other works done in this field. Especially, we investigate differentiability and convexity properties, and present characterizations of the set of solutions to variational inequalities. We then present new descent algorithms for variational inequalities within this framework, including approximate solutions of the direction finding and line search problems. The new class of merit functions include the primal and dual gap functions, introduced by Zuhovickii et al. (1969a, 1969b), and the differentiable merit function recently presented by Fukushima (1992); also, the descent algorithm proposed by Fukushima is a special case from the class of descent methods developed in this paper. Through a generalization of Auchmuty's class of merit functions we extend those inherent in the works of Dafermos (1983), Cohen (1988) and Wu et al. (1991); new algorithmic equivalence results, relating these algorithm classes to each other and to Auchmuty's framework, are also given.