Sparse solutions of linear complementarity problems

This paper considers the characterization and computation of sparse solutions and least-p-norm $$(0<p<1)$$(0<p<1) solutions of the linear complementarity problem $$\hbox {LCP}(q,M)$$LCP(q,M). We show that the number of non-zero entries of any least-p-norm solution of the $$\hbox {LCP}(q,M)$$LCP(q,M) is less than or equal to the rank of M for any arbitrary matrix M and any number $$p\in (0,1)$$p∈(0,1), and there is $$\bar{p}\in (0,1)$$p¯∈(0,1) such that all least-p-norm solutions for $$p\in (0, \bar{p})$$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.