一种超越全局Lipschitz梯度连续性的非单调外推近端梯度-子梯度算法
A nonmonotone extrapolated proximal gradient-subgradient algorithm beyond global Lipschitz gradient continuity
AI总结：
本文提出一种无需问题参数的新型算法，通过精心设计的非单调线搜索和外推步骤，解决非全局Lipschitz梯度连续性问题，并在Kurdyka-Ł ojasiewicz性质下建立收敛分析。
摘要：
随着现代应用的发展，越来越多的复合优化问题出现，其光滑部分不具有全局Lipschitz连续的梯度。这种设定限制了近端梯度（PG）方法及其变体的直接应用，并促使了新的PG型方法及其收敛理论的研究，特别是无需对迭代项施加显式或隐式有界性假设的全局收敛分析。直到最近，首次完整的此类分析才在PG方法及其特定非单调变体中建立，从而激发了进一步的研究。本文考虑了一种超越全局Lipschitz梯度连续性设定的一般复合优化模型。我们提出了一种新颖的问题参数无关算法，结合精心设计的非单调线搜索以处理非全局Lipschitz梯度连续性，以及外推步骤以实现潜在的加速。尽管结合外推与非单调线搜索带来了额外的技术挑战，我们仍在此Kurdyka-Ł ojasiewicz性质下建立了所提算法的细化收敛分析，无需对迭代项施加任何有界性假设。本文因此进一步推进了非全局Lipschitz梯度连续性设定下PG型方法的理论理解。最后，我们进行了数值实验以展示所提算法的有效性，并突出整合外推与非单调线搜索的优势。
英文摘要：
With the advancement of modern applications, an increasing number of composite optimization problems arise whose smooth component does not possess a globally Lipschitz continuous gradient. This setting prevents the direct use of the proximal gradient (PG) method and its variants, and has motivated a growing body of research on new PG-type methods and their convergence theory, in particular, global convergence analysis without imposing any explicit or implicit boundedness assumptions on the iterates. Until recently, the first complete analysis of this kind has been established for the PG method and its specific nonmonotone variants, which has since stimulated further exploration along this research direction. In this paper, we consider a general composite optimization model beyond the global Lipschitz gradient continuity setting. We propose a novel problem-parameter-free algorithm that incorporates a carefully designed nonmonotone line search to handle the non-global Lipschitz gradient continuity, together with an extrapolation step to achieve potential acceleration. Despite the added technical challenges introduced by combining extrapolation with nonmonotone line search, we establish a refined convergence analysis for the proposed algorithm under the Kurdyka-Ł ojasiewicz property, without requiring any boundedness assumptions on the iterates. This work thus further advances the theoretical understanding of PG-type methods in the non-global Lipschitz gradient continuity setting. Finally, we conduct numerical experiments to illustrate the effectiveness of our algorithm and highlight the advantages of integrating extrapolation with a nonmonotone line search.
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