Algorithms For Nonlinear Complementarity Problem And Their Generalization

Complementarity problem is to find a variable or a function that satisfies a certain nonnegative property and complementary relationship which are common exists in our daily life. The related theories about the NCPs have developed and been the most important part of Mathematical Programming since it was proposed first in 1963. In this paper, we focus on the modified and improvement approaches of the nonlinear complementarity problem and generalized problemMany algorithms have been proposed for solving nonlinear complementarity problem. One of the powerful approaches is using merit function to reformulate the nonlinear complementarity problem as an equivalent unconstrained optimization problem or as an equivalent system of nonlinear equations. Many kinds of merit functions appear to lead to a system of smoothing equations and no smoothing equations. Actually, the practical application of these methods do not easily controlled. Based on these shortcomings, the existed methods for nonlinear complementarity problems are further modified in this paper. In order to reduce the calculation scale, we present an improved filter algorithm with SPN decomposition strategy, so that it has better numerical results compared to the method that without the filer technique.Generalized nonlinear complementarity problem is generalized by the NCPs. It is usually reformulated as an equivalent constrained optimization problem to obtain the optimal solutions. But the construct of the objective function is complex. The computation of hessian matrix is difficult to ensure the local superlinerly convergence. So based on the smoothing NCP function, we reformulate the generalized nonlinear complementarity problem over a polyhedral cone as a smoothing system of equations and then propose a new smoothing inexact Newton method for solving it. Under suitable conditions, we show that any accumulation point of the generated sequence is a solution of the problem. We also obtain its global convergence under weaker conditions, and we further establish its local superliner convergence.