Gap Functions And Weak Sharpness Of Solutions For Variational Inequalities

Variational inequality problem is one of classical mathematical problems. Manymodels of problems in physics and engineer are constructed by partial differential equa-tions with some suitable boundary conditions and primal conditions and described bydifferent kinds of variational inequality problems. Its importance is becoming clearer allthe time since it is closely associated with other mathematical branches such as comple-mentary problem, optimization problem, equilibrium problem, fixed point problem, andhas extensive application in science and economics.Researchers have paid much attention to theoretical and numerical analysis of vari-ational inequalities. Recently, many scholars made efforts to formulate variational in-equalities as constrained or unconstrained optimization problem. A real-valued functiondefined on the whole space or its subset is called a gap function, if the set of its globalminimizers on a given set coincides with the set of solutions to the variational in equali-ties. A gap function serves as a useful tool for developing algorithms to solve variationalinequalities and for analyzing their convergence properties. Thus, the study of gap func-tions has become one of the important research subjects in variational inequalities.In this paper, we consider two kinds of generalized variational inequalities andconstruct certain generalized gap functions associated with these problems. In partic-ular, these gap functions are everywhere nonnegative and their zero-sets are preciselysolutions of the related variational inequality problems. Results of error bounds for theunderlying variational inequalities are obtained by using the generalized gap functionsunder the condition that the involved mapping F is g-strongly monotone with respect tothe solution, but not necessarily continuous differentiable, even is not locally Lipschitz.Marcotte and Zhu proposed a kind of gap function which is called dual gap functionand proved that it provide global error bound for classical variational inequality problemis equivalent to the weak sharpness of variational inequalities . Weak sharpness for opti-mization problem plays an important role in sensitivity and convergence of algorithms.In this paper, we introduce the notion of weak sharpness (weak subsharpness) of the solution set for variational inequality problem in a re?exive, strictly convex and smoothBanach space and present its several equivalent conditions. Moreover, we character-ize finite convergence of an arbitrary iterative algorithm under the assumption that thesolution set is weakly subsharp. As a consequence, we show that the proximal pointalgorithm possesses finite convergence.