Igor V. Konnov

Duality for equilibrium problems under generalized monotonicity

Duality is studied for an abstract equilibrium problem which includes, among others, optimization problems and variational inequality problems. Following different schemes, various duals are proposed and primal–dual relationships are established under certain generalized convexity and generalized monotonicity assumptions. In a primal–dual setting, existence results for a solution are derived for different generalized monotone equilibrium problems within each duality scheme.

On variational inequalities for auction market problems

We give an equivalent variational inequality formulation for a general class of equilibrium problems based upon auction decision rules. We show that a general relaxation iterative process with conditional gradient extrapolation ensures convergence to a solution under rather mild assumptions.

A Combined Relaxation Method for a Class of Nonlinear Variational Inequalities

In this paper, we describe a class of combined relaxation methods for the non strictly monotone nonlinear variational inequality problem, which involves a max-type convex function. This method is readily implementable and attains a linear rate of convergence under certain additional assumptions.

The proximal point method for nonmonotone variational inequalities

We consider an application of the proximal point method to variational inequality problems subject to box constraints, whose cost mappings possess order monotonicity properties instead of the usual monotonicity ones. Usually, convergence results of such methods require the additional boundedness assumption of the solutions set. We suggest another approach to obtaining convergence results for proximal point methods which is based on the assumption that the dual variational inequality is solvable. Then the solutions set may be unbounded. We present classes of economic equilibrium problems which satisfy such assumptions.

Partial proximal point method for nonmonotone equilibrium problems

We consider a general equilibrium problem defined on a convex set, whose cost bifunction may be nonmonotone. We show that this problem can be solved by the inexact partial proximal point method. These results can be viewed as a generalization of the known convergence properties of the usual proximal point method.

Lexicographic and sequential equilibrium problems

The aim of this work is to analyze lexicographic equilibrium problems on a topological Hausdorff vector space X, and their relationship with some other vector equilibrium problems. Existence results for the tangled lexicographic problem are proved via the study of a related sequential problem. This approach was already followed by the same authors in the case of variational inequalities.

On Convergence Properties of a Subgradient Method

In this article, we consider convergence properties of the normalized subgradient method which employs the stepsize rule based on a priori knowledge of the optimal value of the cost function. We show that the normalized subgradients possess additional information about the problem under consideration, which can be used for improving convergence rates based on the usual subgradient properties. We also present several convergence results for inexact versions of the method.

An Alternative Economic Equilibrium Model with Different Implementation Mechanisms

We consider a general economic equilibrium model with divisible commodities and price functions based on the material balance condition. We show that mechanisms for attaining its solution points are closely related to information exchange schemes attributed to the model within its basic information framework. In particular, it contains features from both perfect and imperfect competitive models and its equilibrium state can be attained within a completely decentralized transaction mechanism. Therefore, we obtain a rather flexible modeling framework for describing various complex economic systems. At the same time, the model admits an equivalent variational inequality formulation, hence there exists a great collection of results for its investigation and solution. We discuss its relationships with some other economic equilibrium models.

Lexicographic variational inequalities with applications

In this article, we consider equivalence properties between various kinds of lexicographic variational inequalities. By employing various concepts of monotonicity, we show that the usual sequential variational inequality is equivalent to the direct lexicographic variational inequality or to the dual lexicographic variational inequality. We establish several existence results for lexicographic variational inequalities. Also, we introduce the lexicographic complementarity problem and establish its equivalence with the lexicographic variational inequality. We illustrate our approach by several examples of applications to vector transportation and vector spatial equilibrium problems.

A Class of Combined Relaxation Methods for Decomposable Variational Inequalities

Combined relaxation methods are convergent to a solution of variational inequality problems under rather mild assumptions and admit various auxiliary procedures within their two-level structure. In this work, we consider ways to construct decomposition schemes within one class of combined relaxation methods, which maintain useful convergence properties. An application to primal-dual variational inequality problems is also given.