发表机构
Università di Firenze; Sapienza Università di Roma（佛罗伦萨大学; 罗马第一大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究非凸几何条件下非线性优化问题，提出将动量项融入投影梯度法框架的方法，通过合适回溯机制确保收敛，经数值测试，该方法利用动量项信息，在计算上有效。
AI 中文摘要
具有复杂、非凸但几何结构约束的非线性优化问题可通过投影梯度法解决：在弱正则性假设下，这些方法最近被证明具有收敛到最强平稳条件的性质。本文展示了常用于非线性优化以加速收敛过程的动量项如何融入该算法框架而不损害收敛保证。首先强调了在投影方法中直接用一般下降方向替代负梯度所引发的内在问题。接着提出了预投影步骤的合适回溯机制，使我们能在方向中融入动量项。通过该技术，在无任何光滑性假设下，只要基方向对小步长渐近地恢复为负梯度，就能确保收敛到莫尔杜霍维奇平稳性；此外，如果基搜索方向对最小步长恰好恢复为负梯度，该算法被证明分别在有无（局部）光滑性假设下收敛到布里冈和平近端平稳点。最后，在稀疏性和有界秩约束问题等几类问题上对所提过程进行了数值测试；结果表明所提方法利用动量项提供的额外信息在计算上是有效的。
英文摘要
Nonlinear optimization problems with complicated, nonconvex, yet geometrically structured constraints can be tackled by projected-gradient methods: under weak regularity assumptions, these approaches were recently proved to possess convergence properties to the strongest stationarity conditions. In this work, we show how momentum terms, commonly used in nonlinear optimization to speed up the convergence process, can be integrated within this algorithmic framework without harming convergence guarantees. Preliminarily, we highlight an intrinsic issue induced by the direct replacement of the negative gradient with a general descent direction within the projected approach. Then, we present suitable backtracking mechanisms for the pre-projection step, allowing us to integrate momentum terms in the direction. By this technique, we can specifically ensure, without any smoothness assumptions, convergence to Mordukhovich stationarity as long as the base directions asymptotically revert to the negative gradient for small stepsizes; moreover, if the base search direction reverts exactly to the negative gradient for the smallest steps, the algorithm is proved to converge to Bouligand and Proximally stationary points, with and without (local) smoothness assumptions respectively. Finally, the proposed procedure is numerically tested on some classes of problems, namely, sparsity and bounded-rank constrained problems; the results indicate that the proposed method is computationally effective, taking advantage of the additional information provided by the momentum term.
