Newton Hard-Thresholding Pursuit for Sparse Linear Complementarity Problem via A New Merit Function

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.