Two New Algorithms For Variational Inequality

The variational inequality problem (VIP) is an important branch in operationalresearch, which has widely applications in science, engineering and economics. Es-pecially, the numerical methods for VIP attracts many scholars' more attention inrecent years. An important numerical method for VIP is to construct algorithms forthe corresponding KKT conditions.In general, the KKT conditions for VIP is equiva-lently transformed as a system of nonsmooth equations by using Fischer-Burmeisterfunction. Different from the methods in literature, we proposed two new numericalmethods for VIP in the paper as follows:1. At first, a revised Levenberg-Marquardt method was presented for constraintsmooth equations system, which overcome the drawbacks of the parameter in [4]must be a constant number. Same convergent results in [4] were obtained undermild assumptions.2. A new equivalent reformulation of constraint smooth equations system to theKKT conditions for VIP was proposed, and a new Levenberg-Marquardt methodfor solving VIP was constructed by using above mentioned method. The convergentresults were shown under some suitable conditions.3. A new equivalent reformulation of nonlinear equations system without con-straints condition to the KKT conditions for VIP was proposed. It has fewer vari-ables and simpler form, and the algorithm is easily implemented. A damped Newtontype algorithm was presented based on it, which needs not to compute the product ofJacobi matrix as in Levenberg-Marquardt method. Under some certain conditions,the global convergence, local superlinear or quadratical convergence were obtained.Numerical results suggested the method is efficient.