SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A772-A799, January 2021.
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.

中文翻译：

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牛顿阈值法求解稀疏线性互补问题

SIAM科学计算杂志，第43卷，第2期，第A772-A799页，2021年1月。
线性互补问题（LCP）的解决方案在许多应用中自然稀疏，例如双矩阵博弈和投资组合部分问题。尽管稀疏度增加了硬度，但稀疏度使优化速度更快，并且可以进行较大规模的计算。因此，我们考虑了稀疏LCP，研究了其解决方案集的存在性和有界性，并引入了新的价值函数，这使我们能够将问题转化为稀疏约束优化。对于某些选定的参数，该函数证明是连续可微的，并且两次连续可微。有趣的是，如果所涉及的矩阵是正半定的，它也是凸的。然后，我们探索稀疏LCP的解集与稀疏约束优化的固定点之间的关系。最后，采用牛顿硬阈值法求解稀疏约束模型。数值实验表明，该问题可以通过新的价值函数得到有效解决。