发表机构
Southern University of Science and Technology; Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences; University of Chinese Academy of Sciences; National Center for Applied Mathematics Shenzhen（南方科技大学; 中国科学院数学与系统科学研究院计算数学与科学工程计算研究所; 中国科学院大学; 深圳应用数学国家研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对带互补约束的线性和二次规划，提出随机分支方法，利用线性规划子问题求解B-平稳点，实验验证了其在大规模系统上的有效性与竞争性性能。
AI 中文摘要
我们开发了随机分支方法，仅使用线性规划子问题来寻找带互补约束的线性规划（LPCC）和二次规划（QPCC）的Bouligand-平稳（B-平稳）点。这些方法利用可行集的有限并集几何结构，在随机选择的兼容分支上搜索一阶下降。对于LPCC，一种精确方法在采样分支上进行优化，而不精确单纯形法可以在将采样的惩罚问题求解到最优之前接受一个改进的分支可行顶点。对于QPCC，包括具有不定二次目标的问题，分支二次规划被线性化的信赖域子问题所替代；比率测试确保实际下降，阈值化采样检测仅在累积点处出现的分支。在所陈述的假设下，LPCC方法在有限次变化后几乎必然稳定在B-平稳点。对于QPCC方法，有限终止产生一个B-平稳点，并且无限运行中的每个累积点几乎必然是B-平稳的。在双层诱导实例、由逆二次规划产生的实例以及稀疏仿射广义纳什均衡实例上的实验，连同129个MacMPEC嵌入测试，表明这些方法在大型互补系统上返回具有竞争性目标质量和运行时间的点。
英文摘要
We develop randomized branch methods for finding Bouligand-stationary (B-stationary) points of linear and quadratic programs with complementarity constraints (LPCCs and QPCCs) using only linear programming subproblems. The methods exploit the finite-union geometry of the feasible set and search for first-order descent on randomly selected compatible branches. For LPCCs, an exact method optimizes over sampled branches, while an inexact simplex method can accept an improving branch-feasible vertex before solving the sampled penalty problem to optimality. For QPCCs, including problems with indefinite quadratic objectives, branch quadratic programs are replaced by linearized trust-region subproblems; a ratio test ensures actual decrease, and thresholded sampling detects branches that emerge only at accumulation points. Under the stated assumptions, the LPCC methods stabilize after finitely many changes at B-stationary points almost surely. For the QPCC method, finite termination yields a B-stationary point, and every accumulation point of an infinite run is B-stationary almost surely. Experiments on bilevel-induced instances, instances arising from inverse quadratic programming, and sparse affine generalized Nash equilibrium instances, together with 129 MacMPEC embedding tests, show that the methods return points with competitive objective quality and runtimes on large-scale complementarity systems.
Comments46 pages, 2 figures
