Nonsmooth Projection-Free Optimization with Functional Constraints
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914930478809088 |
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| author | Asgari, Kamiar Neely, Michael J. |
| author_facet | Asgari, Kamiar Neely, Michael J. |
| contents | This paper presents a subgradient-based algorithm for constrained nonsmooth convex optimization that does not require projections onto the feasible set. While the well-established Frank-Wolfe algorithm and its variants already avoid projections, they are primarily designed for smooth objective functions. In contrast, our proposed algorithm can handle nonsmooth problems with general convex functional inequality constraints. It achieves an $ε$-suboptimal solution in $\mathcal{O}(ε^{-2})$ iterations, with each iteration requiring only a single (potentially inexact) Linear Minimization Oracle (LMO) call and a (possibly inexact) subgradient computation. This performance is consistent with existing lower bounds. Similar performance is observed when deterministic subgradients are replaced with stochastic subgradients. In the special case where there are no functional inequality constraints, our algorithm competes favorably with a recent nonsmooth projection-free method designed for constraint-free problems. Our approach utilizes a simple separation scheme in conjunction with a new Lagrange multiplier update rule. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_11180 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonsmooth Projection-Free Optimization with Functional Constraints Asgari, Kamiar Neely, Michael J. Optimization and Control Machine Learning Systems and Control 65K05, 65K10, 65K99, 90C25, 90C15, 90C25, 90C30 This paper presents a subgradient-based algorithm for constrained nonsmooth convex optimization that does not require projections onto the feasible set. While the well-established Frank-Wolfe algorithm and its variants already avoid projections, they are primarily designed for smooth objective functions. In contrast, our proposed algorithm can handle nonsmooth problems with general convex functional inequality constraints. It achieves an $ε$-suboptimal solution in $\mathcal{O}(ε^{-2})$ iterations, with each iteration requiring only a single (potentially inexact) Linear Minimization Oracle (LMO) call and a (possibly inexact) subgradient computation. This performance is consistent with existing lower bounds. Similar performance is observed when deterministic subgradients are replaced with stochastic subgradients. In the special case where there are no functional inequality constraints, our algorithm competes favorably with a recent nonsmooth projection-free method designed for constraint-free problems. Our approach utilizes a simple separation scheme in conjunction with a new Lagrange multiplier update rule. |
| title | Nonsmooth Projection-Free Optimization with Functional Constraints |
| topic | Optimization and Control Machine Learning Systems and Control 65K05, 65K10, 65K99, 90C25, 90C15, 90C25, 90C30 |
| url | https://arxiv.org/abs/2311.11180 |