Sparse solutions of linear complementarity problems

This paper considers the characterization and computation of sparse solutions and least-p-norm (0 < p< 1) solutions of the linear complementarity problem LCP (q, M). We show that the number of non-zero entries of any least-p-norm solution of the LCP (q, M) is less than or equal to the rank of M for any arbitrary matrix M and any number p∈ (0 , 1) , and there is p¯ ∈ (0 , 1) such that all least-p-norm solutions for p∈ (0 , p¯) are sparse solutions. Moreover, we provide conditions on M such that a sparse solution can be found by solving convex minimization. Applications to the problem of portfolio selection within the Markowitz mean-variance framework are discussed.