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Using vector divisions in solving the linear complementarity problem. (English) Zbl 1238.65053
Summary: The linear complementarity problem LCP(M,q) is to find a vector z in n satisfying z T (Mz+q)=0, Mz+q0, z0, where M=(m ij ) n×n and q n are given. In this paper, we use the fact that solving LCP(M,q) is equivalent to solving the nonlinear equation F(x)=0 where F is a function from n into itself defined by F(x)=(M+I)x+(M-I)x+(M-I)|x|+q. We build a sequence of smooth functions F ˜(p,x) which is uniformly convergent to the function F(x). We show that, an approximation of the solution of the LCP(M,q) (when it exists) is obtained by solving F ˜(p,x)=0 for a parameter p large enough. Then we give a globally convergent hybrid algorithm which is based on vector divisions and the secant method for solving LCP(M,q). We close our paper with some numerical simulations to illustrate our theoretical results, and to show that this method can solve efficiently large-scale linear complementarity problems.
65K05Mathematical programming (numerical methods)
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
65H10Systems of nonlinear equations (numerical methods)
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