Lasry-Lions Envelopes and Nonconvex Optimization: A Homotopy Approach
This work addresses optimization problems in domains like signal processing and data analysis where nonsmooth and nonconvex terms are common, offering a potentially more efficient approach, though it appears incremental as it builds on existing envelope techniques.
The paper tackles the challenge of optimizing nonsmooth and nonconvex functions by using Lasry-Lions double envelopes to approximate these terms, enabling the development of a homotopy method that builds easier-to-solve subproblems. It demonstrates the method's effectiveness in signal decoding and spectral unmixing, showing it can outperform classical alternatives in certain settings.
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.