A Constrained Optimization Approach for LCP (Journal of Computational Mathematics, 2004)
In this paper, LCP is converted to an equivalent nonsmooth nonlinear equation system $H(x, y)=0$ by using the famous NCP function–Fischer-Burmeister function. Note that some equations in $H(x, y)=0$ are nonsmooth and nonlinear hence difficult to solve while the others are linear hence easy to solve. Then we further convert the nonlinear equation system $H(x, y)=0$ to an optimization problem with linear equality constraints. After that we study the conditions under which the K–T points of the optimization problem are the solutions of the original LCP and propose a method to solve the optimization problem. In this algorithm, the search direction is obtained by solving a strict convex programming at each iterative point. However, our algorithm is essentially different from traditional SQP method. The global convergence of the method is proved under mild conditions. In addition, we can prove that the algorithm is convergent superlinearly under the conditions: $M$ is $P_0$ matrix and the limit point is a strict complementarity solution of LCP. Preliminary numerical experiments are reported with this method.
Keywords: LCP; Strict complementarity; Nonsmooth equation system; $P_0$ matrix; Superlinear convergence.