Gap functions and error bounds for random generalized variational inequality problems

Suhel Ahmad Khan,Javid Iqbal,Yekini Shehu

doi:10.1186/s13660-016-0984-5

Suhel Ahmad Khan, Javid Iqbal + Show 1 more

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https://doi.org/10.1186/s13660-016-0984-5

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Abstract

This paper is devoted to the study of gap functions of random generalized variational inequality problems in a fuzzy environment. Further by using the residual vector we compute error bounds for random generalized variational inequality problems and we generalize the regularized gap functions proposed by Fukushima. Furthermore we study various properties of the generalized regularized gap functions in random fuzzy mappings and derive the global error bounds for random generalized variational inequality problems. Our results are new and generalize a number of known results to generalized variational inequality problems with fuzzy mappings.

Highlights

1 Introduction The theory of gap function was introduced for the study of a convex optimization problem and subsequently applied to variational inequality problems
We show that Rφθ(t)(t, x(t)) plays the role of natural residual vector in random fuzzy mapping for the RGVIP ( . )
4 Conclusion In this paper, the concept of gap functions for random generalized variational inequality problems has been introduced by using the fuzzy residual vector

Summary

Introduction

The theory of gap function was introduced for the study of a convex optimization problem and subsequently applied to variational inequality problems. One of the classical approaches in the analysis of a variational inequality problem is to transform it into an equivalent optimization problem via the notion of a gap function. Some effort has been made to develop gap functions for various classes of variational inequality problems; see for example [ – ]. Variational inequalities in the setting of fuzzy mappings have been introduced and studied which are closely related with fuzzy optimization and decision-making problems. Many efforts have been made to reformulate the variational inequality problems and optimization problems in the fuzzy mapping. Variational inequality problems have been generalized and extended in various directions using novel techniques of fuzzy theory

Objectives

Conclusion