Sparse solutions of linear complementarity problems

This paper considers the characterization and computation of sparse solutions and leastp-norm (0 < p < 1) solutions of the linear complementarity problems 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 p̄ ∈ (0, 1) such that all least-p-norm solutions for p ∈ (0, p̄) 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.