Saddle points and gap functions for vector equilibrium problems via conjugate duality in vector optimization

In this paper, two conjugate dual problems based on weak efficiency to a constrained vector optimization problem are introduced. Some inclusion relations between the dual objective mappings and the properties of the Lagrangian maps and their saddle points for primal problem are discussed. Gap functions for a vector equilibrium problem are established by using the weak and strong duality.     

Gap functions for generalized vector equilibrium problems via conjugate duality and applications

In this article, gap functions for a generalized vector equilibrium problem (GVEP) with explicit constraints are investigated. Under a concept of supremum/infimum of a set, defined in terms of a closure of the set, three kinds of conjugate dual problems are investigated by considering the different perturbations to GVEP. Then, gap functions for GVEP are established by using the weak and strong duality results. As application, the proposed approach is applied to construct gap functions for a vector optimization problem and a generalized vector variational inequality problem.     

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.Corresponding author.     

Regularized gap functions for variational problems

We consider variational problems in Banach spaces. Well-posedness concepts for such problems are introduced and investigated by means of two gap functions and their Moreau-Yosida regularizations.     

Weak sharpness for gap functions in vector variational inequalities

In this paper, characterizations of the set of solutions for VVI are presented by using scalarization approaches. The set of solutions of VVI is shown to be the set of weak sharpness for gap functions of some scalarization of VVI and for gap functions of VVI under semi-strong monotonicity. Some examples are given to illustrate these results.     

Iterative methods for solving monotone equilibrium problems via dual gap functions

This paper proposes an iterative method for solving strongly monotone equilibrium problems by using gap functions combined with double projection-type mappings. Global convergence of the proposed algorithm is proved and its complexity is estimated. This algorithm is then coupled with the proximal point method to generate a new algorithm for solving monotone equilibrium problems. A class of linear equilibrium problems is investigated and numerical examples are implemented to verify our algorithms.     

Regularized gap functions for nonsmooth variational inequality problems

Under the condition that the involved function F is locally Lipschitz, but not necessarily differentiable, we investigate the regularized gap function defined by a generalized distance function for the variational inequality problem (VIP). First, we compute exactly the Clarke-Rockafellar directional derivatives of the regularized gap functions (and of some modified ones). Second, using these results, we show that, under the strongly monotonicity assumption, the regularized gap functions have fractional exponent error bounds, and thereby we provide an algorithm of Armijo type to solve the VIP.     

Some equilibrium problems under uncertainty and random variational inequalities

In this paper we describe some nonlinear equilibrium problems under uncertainty arising from economics and operations research. In particular we treat Wardrop equilibria in traffic networks. We show how the theory of monotone random variational inequalities, where random variables occur both in the operator and the constraint set, can be applied to model these problems. Therefore in this contribution we introduce the topic of random variational inequalities and present some of our recent results in this field. In particular, we treat the more structured case where a finite Karhunen-Loève expansion leads to a separation of the random and the deterministic variables. Here we describe a norm convergent approximation procedure based on averaging and truncation. We illustrate this procedure by means of some small sized numerical examples.     

Modified golden ratio algorithms for pseudomonotone equilibrium problems and variational inequalities

Applications of Mathematics - We propose a modification of the golden ratio algorithm for solving pseudomonotone equilibrium problems with a Lipschitz-type condition in Hilbert spaces. A new...     

Differential and sensitivity properties of gap functions for Minty vector variational inequalities

The purpose of this paper is to investigate differential properties of a class of set-valued maps and gap functions involving Minty vector variational inequalities. Relationships between their contingent derivatives are discussed. An explicit expression of the contingent derivative for the class of set-valued maps is established. Optimality conditions of solutions for Minty vector variational inequalities are obtained.     

Second-order duality for the variational problems

A dual problem associated with a class of variational problems is formulated that involves second derivatives of the functions. Under the invexity assumptions on the functions that compose the primal problems, second-order duality results (weak duality, strong duality and converse duality) are derived for this pair of problems.     

A network formulation of market equilibrium problems and variational inequalities

This paper shows that market equilibrium problems of production may generally be modelled as equilibrium flow problems in networks and that their equilibrium conditions can be visualized as a variational inequality. This connection would allow us to transplant directly elements of the well-developed theory of equilibrium flow in networks to the theory of market equilibrium.     

Existence theorems via duality for equilibrium problems with trifunctions

We present an existence result for an equilibrium problem formulated with trifunctions, which is motivated by variational inequalities governed by quasimonotone operators. To prove the existence result, we define the dual problem, and some monotonicity notions for trifunctions. From the main result follow, among others, the Browder–Minty theorem for variational inequalities and Ky Fan’s Minimax theorem. Some applications for mixed equilibrium problems and variational inequalities are given.     

A generalized duality method for solving variational inequalities Applications to some nonlinear Dirichlet problems

Summary. In [2] Bermúdez and Moreno introduced a duality algorithm for the numerical solution of variational inequalities; this algorithm is based on some properties of the Yosida regularization of maximal monotone operators. The performances of this algorithm strongly depend on the choice of two constant parameters. A generalization of the algorithm with automatic choice of parameters was discussed in [13], where the constant parameters were replaced by scalar functions, thus improving the convergence of the algorithm. In this article we present a generalization of the Bermúdez-Moreno algorithm that allows the use of very general operators as parameters, extending some of the results in [2], [13] and [14]. As a particular case, we analyze the use of scalar and matrix-valued parameters in a L^p()^M context. We apply the results developed to some boundary value problems involving the p-Laplacian operator, where it is shown that the use of matrix-valued parameters improves the convergence of the algorithm.Mathematics Subject Classification (1991): 47J20 – 65N12     

Revisiting subgradient extragradient methods for solving variational inequalities

Numerical Algorithms - In this paper, several extragradient algorithms with inertial effects and adaptive non-monotonic step sizes are proposed to solve pseudomonotone variational inequalities in...     

Farkas-type results for inequality systems with composed convex functions via conjugate duality

We present some Farkas-type results for inequality systems involving finitely many functions. Therefore we use a conjugate duality approach applied to an optimization problem with a composed convex objective function and convex inequality constraints. Some recently obtained results are rediscovered as special cases of our main result.     

Nonconvex functions and variational inequalities

In this paper, we study some properties of a class of nonconvex functions, called semipreinvex functions, which includes the classes of preinvex functions and arc-connected convex functions. It is shown that the minimum of an arcwise directionally differentiable semi-invex functions on a semi-invex set can be characterized by a class of variational inequalities, known as variational-like inequalities. We use the auxiliary principle technique to prove the existence of a solution of a variational-like inequality and suggest a novel iterative algorithm.     

Sensitivity analysis of gap functions for vector variational inequality via coderivatives

The aim of this article is to investigate codifferential properties of a class of set-valued maps and gap function involving vector variational inequality. Relationships between their coderivatives are discussed. Formulae for computing coderivatives of the gap function are established. Optimality conditions of solutions for vector variational inequalities are obtained. The finite-dimensional cases are also discussed.