  Solutions to the linear complementarity problem (LCP) are naturally sparse in
many applications such as bimatrix games and portfolio section problems.
Despite that it gives rise to the hardness, sparsity makes optimization faster
and enables relatively large scale computation. Motivated by this, we take the
sparse LCP into consideration, investigating the existence and boundedness of
its solution set as well as introducing a new merit function, which allows us
to convert the problem into a sparsity constrained optimization. The function
turns out to be continuously differentiable and twice continuously
differentiable for some chosen parameters. Interestingly, it is also convex if
the involved matrix is positive semidefinite. We then explore the relationship
between the solution set to the sparse LCP and stationary points of the
sparsity constrained optimization. Finally, Newton hard thresholding pursuit is
adopted to solve the sparsity constrained model. Numerical experiments
demonstrate that the problem can be efficiently solved through the new merit
function.
