Some Reseach On The Gap Functions And Algorithm And Complementarity Problem Of A Kind Of Variational Inequality

Variational inequality problem is one of classical mathematical problems． It iswidely applied to some application field，such as mechanics，differential equations，cybernetics， mathematical economics， countermeasure theory， optimizationtheory， nonlinear programming． Variational inequality and its application areconcerned by scholars at home and broad day by day， since it has intimateconnection between variational inequality problem and complementarity problem，optimization problem， equilibrium problem， fixed point theory．In this paper，we consider a kind of variational inequalities and construct certaingeneralized gap functions associated with algorithm problems． We do some work asfollows:In the first chapter， we give the introduction of the recent work of many authorsin this field in domestic and abroad．In chapter two， we consider a class of set-valued mixed variational inequalitieswith its certain generalized gap functions and generalized D-gap functions in realHilbert spaces． And we give the global error bounds of this kind variationalinequalities through those two generalized gap functions．In chapter three， on the basis of chapter two， we equivalently transform thegeneralized variational inequalities into constraint optimization problem through gapfunction thoughte．And we gave its the steepest descent method and its convergence．In chapter four， we discuss the nonlinear complementarity problems accordingto the neural network density function． The smooth Newton method of the nonlinearcomplementarity problems and its convergence are obtained．...