Lasry-Lions包络与非凸优化：一种同伦方法
Lasry-Lions Envelopes and Nonconvex Optimization: A Homotopy Approach
AI总结：
本文利用Lasry-Lions双包络兼具光滑性与凸化过渡的特点，提出构建渐进近似子问题的同伦优化方法，并分析复合最小化的收敛性及在信号解码和光谱解混中的优势。
摘要：
在大规模优化中，给定问题中存在非光滑且非凸的项，通常会使其难以求解。在凸优化中，处理非光滑项的一种流行方法是用它们各自的Moreau包络来近似这些项。在本工作中，我们研究使用Lasry-Lions双包络来近似同样非凸的非光滑项。这些包络是Moreau包络的扩展，但表现出额外的光滑性，使其适用于快速优化算法。Lasry-Lions包络也可以被视为给定函数与其凸包络之间的“中间物”，我们利用这一性质开发了一种方法，该方法构建一系列近似子问题，而这些子问题比原始问题更容易求解。我们讨论了该方法用于处理复合最小化问题时的收敛性质；此外，基于多项实验，我们讨论了在两个领域中它可能比经典替代方案更有用的情形：信号解码和光谱解混。
英文摘要：
In large-scale optimization, the presence of nonsmooth and nonconvex terms in a given problem typically makes it hard to solve. A popular approach to address nonsmooth terms in convex optimization is to approximate them with their respective Moreau envelopes. In this work, we study the use of Lasry-Lions double envelopes to approximate nonsmooth terms that are also not convex. These envelopes are an extension of the Moreau ones but exhibit an additional smoothness property that makes them amenable to fast optimization algorithms. Lasry-Lions envelopes can also be seen as an "intermediate" between a given function and its convex envelope, and we make use of this property to develop a method that builds a sequence of approximate subproblems that are easier to solve than the original problem. We discuss convergence properties of this method when used to address composite minimization problems; additionally, based on a number of experiments, we discuss settings where it may be more useful than classical alternatives in two domains: signal decoding and spectral unmixing.
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