文献类型：期刊文献

英文题名：Approximation Methods for a Class of Non-Lipschitz Mathematical Programs with Equilibrium Constraints

作者：Guo, Lei[1];Li, Gaoxi[2]

机构：[1]East China Univ Sci & Technol, Sch Business, Shanghai 200237, Peoples R China;[2]Chongqing Technol & Business Univ, Sch Math & Stat, Chongqing 400067, Peoples R China

年份：2024

卷号：202

期号：3

起止页码：1421

外文期刊名：JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS

收录：;EI(收录号：20242716606976);WOS:【SCI-EXPANDED(收录号:WOS:001259362100001)】；

基金：The authors are grateful to the two referees for their helpful comments and constructive suggestions. In particular, we thank one of the referees for suggesting the use of weaker qualifications than MPEC-RCPLDQC when studying the convergence. The first author was supported by the National Natural Science Foundation of China (Grants 72131007, 72140006, 12271161) and the Natural Science Foundation of Shanghai (Grant 22ZR1415900). This second author was supported by the Project of National Center for Applied Mathematics (Grant ncamc2021-msxm01) and the National Natural Science Foundation of China (Grant 11901068).

语种：英文

外文关键词：Mathematical program with equilibrium constraints; Nonconvex optimization; Non-Lipschitz continuity; Sparsity-inducing penalty; Smoothing function; Regularization method

摘要：We consider how to solve a class of non-Lipschitz mathematical programs with equilibrium constraints (MPEC) where the objective function involves a non-Lipschitz sparsity-inducing function and other functions are smooth. Solving the non-Lipschitz MPEC is highly challenging since the standard constraint qualifications fail due to the existence of equilibrium constraints and the subdifferential of the objective function is unbounded due to the existence of the non-Lipschitz function. On the one hand, for tackling the non-Lipschitzness of the objective function, we introduce a novel class of locally Lipschitz approximation functions that consolidate and unify a diverse range of existing smoothing techniques for the non-Lipschitz function. On the other hand, we use the Kanzow and Schwartz regularization scheme to approximate the equilibrium constraints since this regularization can preserve certain perpendicular structure as in equilibrium constraints, which can induce better convergence results. Then an approximation method is proposed for solving the non-Lipschitz MPEC and its convergence is established under weak conditions. In contrast with existing results, the proposed method can converge to a better stationary point under weaker qualification conditions. Finally, a computational study on the sparse solutions of linear complementarity problems is presented. The numerical results demonstrate the effectiveness of the proposed method.