Lasry-Lions Envelopes and Nonconvex Optimization: A Homotopy Approach
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866929314285486080 |
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| author | Simões, Miguel Themelis, Andreas Patrinos, Panagiotis |
| author_facet | Simões, Miguel Themelis, Andreas Patrinos, Panagiotis |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_08533 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Lasry-Lions Envelopes and Nonconvex Optimization: A Homotopy Approach Simões, Miguel Themelis, Andreas Patrinos, Panagiotis Optimization and Control Computer Vision and Pattern Recognition Signal Processing Machine Learning 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. |
| title | Lasry-Lions Envelopes and Nonconvex Optimization: A Homotopy Approach |
| topic | Optimization and Control Computer Vision and Pattern Recognition Signal Processing Machine Learning |
| url | https://arxiv.org/abs/2103.08533 |