发表机构
University of Würzburg（维尔茨堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对非空闭集上的非光滑非凸优化问题，提出带两种全局化策略的投影次梯度方法，证明其收敛到更强平稳性解，并在MPEC式问题、MAXCUT和Robust PCA上取得良好数值结果。
AI 中文摘要
我们研究在非空闭集（该集不一定是凸集）上最小化满足非光滑版下降引理的非光滑函数的优化问题。目标函数属于上$\boldsymbol{\text{C}}^2$函数类，而约束条件可促进稀疏或低秩结构。我们提出一种带有两种不同全局化策略的投影次梯度方法：(a) 非单调线搜索，以及在额外假设下的(b) 自调方法，其步长由依赖于过去迭代数据的公式给出。我们证明两种方法均收敛到满足比次微分和规则所预期更强的平稳性概念的解，这一点尤为重要，因为所关注的优化问题本质上是非凸的。最后，我们展示将该算法应用于MPEC式问题以及矩阵优化问题MAXCUT和鲁棒主成分分析（Robust PCA）时的良好数值结果。
英文摘要
We investigate the optimization problem of minimizing a nonsmooth function that satisfies a nonsmooth version of the descent lemma over a nonempty and closed but not necessarily convex set. The objective function belongs to the class of upper-$\mathcal{C}^2$ functions, whereas the constraints may promote a sparse or low-rank structure. We propose a projected subgradient method with two different globalization strategies: (a) a nonmonotone linesearch and, under additional assumptions, (b) an auto-conditioned method, where the stepsize is given by a formula depending on data from past iterations. We show that both methods converge to solutions that satisfy a stronger stationarity concept than one would expect from the subdifferential sum-rule, which is particularly important since the optimization problems of interest are inherently nonconvex. Finally, we present promising numerical results when applying the algorithm to an MPEC-style problem as well as the matrix optimization problems MAXCUT and Robust PCA.
