AI 中文总结
研究无约束集值优化问题，提出带和不带线搜索的近端梯度方法，借助向量优化问题弱有效点刻画弱极小点，建立平稳性条件，分析计算复杂度与收敛速度，通过数值结果测试方法性能。
AI 中文摘要
本文提出了两种不同类型的近端梯度方法（带线搜索和不带线搜索），用于解决在由凸、尖且封闭的实锥诱导的下集序关系下的无约束集值优化问题。问题的目标映射涉及有限多个函数，每个函数是连续可微函数与适当且封闭的凸函数之和。借助一族向量优化问题的弱有效点来刻画该问题的弱极小点，建立平稳性条件及其与所研究问题弱极小点的联系。讨论非平稳点处下降方向的概念，为基于线搜索的方法制定 Armijo 型线搜索条件并证明步长的存在性。在温和假设下建立了所提方法的全局收敛性，带线搜索的近端梯度方法的收敛分析在理论上优于先前为集值优化问题中最速下降法建立的收敛结果。此外，分析了所提方法的计算复杂度，表明两种方法都达到了\(\mathcal{O}(1/\sqrt{k})\)的收敛速度，并报告了数值结果以测试方法在实际中的性能。
英文摘要
This work presents two different types of proximal gradient methods, with line search and without line search, for solving unconstrained set-valued optimization problems under the lower set-less ordering relation induced by a solid cone that is convex, pointed, and closed. The objective mapping of the problem involves finitely many functions, with each one being the sum of a continuously differentiable function and a convex function that is proper and closed. We present an approach to characterize weakly minimal points of the problem with the help of weakly efficient points of a family of vector optimization problems. Thereafter, we establish a stationarity condition along with its connection with weakly minimal points of the problem under study. Based on the stationary condition, the concept of a descent direction at a non-stationary point is discussed. In view of the line search-based method, we formulate an Armijo-type line search condition and establish the existence of such a step-size. For the proposed methods, global convergence is established under mild assumptions. The convergence analysis of the proximal gradient method with line search provides a theoretical advancement over the convergence results previously established for the steepest descent method in set-valued optimization problems. In addition, we analyze the computational complexity of the proposed methods and show that both methods achieve a convergence rate of $\mathcal{O}(1/\sqrt{k})$. Numerical results are reported to test the performance of the methods in practice.
