发表机构
Isfahan University of Technology; University of Tehran（伊斯法罕理工大学; 德黑兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出一种投影下降方法，用于在多面体集上最小化弱半光滑函数，通过Clarke ε-次微分近似和线搜索策略，在温和假设下保证收敛，并在图像去噪等应用中验证了有效性。
AI 中文摘要
本研究开发了一种基于投影的下降方法，用于在多面体集上最小化弱半光滑函数，迈出了将基于投影的下降算法扩展到非光滑、非凸优化问题的第一步。在每次迭代中，该方法在当前迭代点处考虑可行域上的Clarke ε-次微分的内部近似，并使用其最小范数元素生成搜索方向。随后开发了一种指数有限回溯线搜索来评估所生成方向的质量，同时提出了Mifflin线搜索的投影变体，以在必要时识别丰富Clarke ε-次微分近似的非冗余次梯度。为了量化平稳性，基于投影算子的平稳点特征描述为该提议方法提供了一种可计算的最优性度量。在温和假设下建立了所提议方法的收敛性质。如果算法生成无限多个严肃步骤，我们识别出两个子序列，使得每个子序列的每个聚点都是平稳的。如果严肃步骤的数量有限，我们表明最终的严肃步骤生成一个平稳点。数值实验证明了所提议方法在广泛测试问题中的效率和广泛适用性，包括图像去噪、多目标优化和数据聚类等应用。为了将所提议方法的适用性扩展到具有一般光滑非线性约束的问题，我们提出了一种基于顺序线性化的启发式方法。
英文摘要
This study develops a projection-based descent method for minimizing weakly semismooth functions over polyhedral sets, taking an initial step toward extending projection-based descent algorithms to nonsmooth, nonconvex optimization problems. At each iteration, the method considers an inner approximation of the Clarke $\varepsilon$-subdifferential at the current iterate over the feasible region and uses its least-norm element to generate a search direction. An exponential limited backtracking line search is then developed to assess the quality of the generated direction, while a projected variant of Mifflin's line search is proposed to identify nonredundant subgradients that enrich the Clarke $\varepsilon$-subdifferential approximation when necessary. To quantify stationarity, a characterization of stationary points based on the projection operator yields a computable optimality measure for the proposed method. The convergence properties of the proposed method are established under mild assumptions. If the algorithm generates infinitely many serious steps, we identify two subsequences such that every cluster point of each subsequence is stationary. If the number of serious steps is finite, we show that the final serious step generates a stationary point. Numerical experiments demonstrate the efficiency and broad applicability of the proposed method across a wide range of test problems, including applications in image denoising, multiobjective optimization, and data clustering. To extend the applicability of the proposed method to problems with general smooth nonlinear constraints, we propose a heuristic approach based on sequential linearization.
