Implicit solution function of P0 and Z matrix linear complementarity constraints

Using the least element solution of the P0 and Z matrix linear complementarity problem (LCP), we define an implicit solution function for linear complementarity constraints (LCC). We show that the sequence of solution functions defined by the unique solution of the regularized LCP is monotonically increasing and converges to the implicit solution function as the regularization parameter goes down to zero. Moreover, each component of the implicit solution function is convex. We find that the solution set of the irreducible P0 and Z matrix LCP can be represented by the least element solution and a Perron-Frobenius eigenvector. These results are applied to convex reformulation of mathematical programs with P0 and Z matrix LCC. Preliminary numerical results show the effectiveness and the efficiency of the reformulation.