Study The Method For Solving Variational Inequality Via Merit Function

In this paper，the main content of the thesis is presented as follows:In Chapter2, Variational inequality problem was reformulated as a constrainedoptimization problem. In this paper，based on this reformulation, a larger non-monotone-stepsearch algorithm based on larger non-monotone-step line search technique is proposed forsolving variational inequality problems. The global convergence properties of the newmethod are proved. The algorithm avoids solving the gradient problem. Numerical resultsshow that the new algorithms are efficient.In Chapter3, Nonlinear complementarity problem was reformulated as a nonnegativeconstrained optimization problem. In this paper, based on this reformulation, a largernon-monotone-step line search algorithm based on larger non-monotone-step line searchtechnique is proposed for solving nonlinear complementarity problems. Under somereasonable conditions, it is proved that the algorithm is globally convergent. Thus, thealgorithm avoids solving the gradient problem. Numerical results show that the newalgorithms are efficient and suitable for solving large-scale problems.In Chapter4, a hybrid Newton method based on Zhang H.C.’s non-monotone line searchtechnique is proposed for minimizing the generalized D-gap function. It is shown that thealgorithm has nice convergence properties. Under some reasonable conditions, it is provedthat the algorithm is globally quadratically convergent. Moreover, when the mapping F isstrongly monotone without Lipschitz condition, it is proved that the generalized D-gapfunction has a global error bound. Numerical results show that the new algorithms areefficient.In Chapter5, we propose a generalized gap function in order to reformulate a class ofgeneralized variational inequality problem(GVIP) as a constrained optimization, based on thisreformulation, a non-monotone algorithm based on Zhang H.C.’s non-monotone-step line search technique is proposed for solving this optimization problems. Under some reasonableconditions, it is proved that the algorithm is globally convergent.