论文标题
AN-SPS：自适应样本量非单身线线搜索光谱投射的亚级别级别方法，用于凸的约束优化问题
AN-SPS: Adaptive Sample Size Nonmonotone Line Search Spectral Projected Subgradient Method for Convex Constrained Optimization Problems
论文作者
论文摘要
我们考虑以数学期望的形式使用可能非平滑目标函数的凸优化问题。所提出的框架（AN-SP）采用样本平均近似值（SAA）来近似目标函数，这是不可用的或太昂贵的计算。样本量以自适应方式选择，最终将SAA误差推到零几乎肯定（A.S.）。搜索方向基于缩放的亚速度和光谱系数，这均与SAA函数有关。步骤大小是通过在预定义的间隔上通过非单调线搜索获得的，该间隔在理论上声音和实际上有效的算法。该方法通过将结果点投影到可行的集合中来保留可行性。 A.S. AN-SPS方法的收敛性被证明没有假设可行的集合或有界的迭代。有关铰链损失问题的初步数值结果揭示了所提出的适应性方案的优势。此外，还对不同的非单调线搜索策略以及在AN-SPS框架内的不同光谱系数结合进行了研究，为将来的工作提供了一些提示。
We consider convex optimization problems with a possibly nonsmooth objective function in the form of a mathematical expectation. The proposed framework (AN-SPS) employs Sample Average Approximations (SAA) to approximate the objective function, which is either unavailable or too costly to compute. The sample size is chosen in an adaptive manner, which eventually pushes the SAA error to zero almost surely (a.s.). The search direction is based on a scaled subgradient and a spectral coefficient, both related to the SAA function. The step size is obtained via a nonmonotone line search over a predefined interval, which yields a theoretically sound and practically efficient algorithm. The method retains feasibility by projecting the resulting points onto a feasible set. The a.s. convergence of AN-SPS method is proved without the assumption of a bounded feasible set or bounded iterates. Preliminary numerical results on Hinge loss problems reveal the advantages of the proposed adaptive scheme. In addition, a study of different nonmonotone line search strategies in combination with different spectral coefficients within AN-SPS framework is also conducted, yielding some hints for future work.
