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.
翻译:在大规模优化中,在特定问题中存在非moot和非convex术语通常很难解决。在配置优化中处理非moot术语的流行方法是将其与各自的Moreau信封相近。在这项工作中,我们研究使用Lasry-Lion双包以近似非moot用词,这些非moot用词也不是 convex。这些信封是莫罗语的延伸,但又表现出一种更顺畅的属性,使其适合快速优化算法。激光-Lion信封也可以被视为在给定函数及其配置信封之间“中间”的“中间”,我们利用这一属性开发出一种方法,建立比原始问题更容易解决的近似子问题序列。我们在使用这种方法解决综合最小化问题时,我们讨论该方法的趋同特性;此外,根据一些实验,我们讨论在两个领域(信号解码和光谱解密)中可能比经典替代方法更有用的环境。