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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2210.05108 |
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| _version_ | 1866918491205926912 |
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| author | Cheng, Yi Lan, Guanghui Masiha, Saeed Romeijn, H. Edwin |
| author_facet | Cheng, Yi Lan, Guanghui Masiha, Saeed Romeijn, H. Edwin |
| contents | We study projection-free methods for functional constrained optimization with convex or smooth nonconvex objectives. Such problems arise in applications such as portfolio optimization and radiation therapy planning, where risk-aware criteria and sparsity frequently appear together. For the convex setting, we propose a Level Conditional Gradient (LCG) method that combines a level-set outer loop with a conditional gradient oracle for saddle-point subproblems, and we show an iteration complexity of $\mathcal{O}\big(ε^{-2}\log(ε^{-1})\big)$ for smooth and nonsmooth cases without dependence on the magnitude of an optimal dual Lagrange multiplier. For the nonconvex setting, we propose the Inexact Proximal Point LCG (IPP-LCG) method, which solves a sequence of convex subproblems by LCG and attains $\mathcal{O}\big(ε^{-3}\log(ε^{-1})\big)$ complexity for computing an \((ε,ε)\)-near-KKT point. Numerical results on portfolio selection and IMRT illustrate the practical sparsity/risk trade-offs of the proposed methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_05108 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Projection-Free Functional Constrained Optimization for Risk Aversion and Sparsity Control Cheng, Yi Lan, Guanghui Masiha, Saeed Romeijn, H. Edwin Optimization and Control Machine Learning We study projection-free methods for functional constrained optimization with convex or smooth nonconvex objectives. Such problems arise in applications such as portfolio optimization and radiation therapy planning, where risk-aware criteria and sparsity frequently appear together. For the convex setting, we propose a Level Conditional Gradient (LCG) method that combines a level-set outer loop with a conditional gradient oracle for saddle-point subproblems, and we show an iteration complexity of $\mathcal{O}\big(ε^{-2}\log(ε^{-1})\big)$ for smooth and nonsmooth cases without dependence on the magnitude of an optimal dual Lagrange multiplier. For the nonconvex setting, we propose the Inexact Proximal Point LCG (IPP-LCG) method, which solves a sequence of convex subproblems by LCG and attains $\mathcal{O}\big(ε^{-3}\log(ε^{-1})\big)$ complexity for computing an \((ε,ε)\)-near-KKT point. Numerical results on portfolio selection and IMRT illustrate the practical sparsity/risk trade-offs of the proposed methods. |
| title | Projection-Free Functional Constrained Optimization for Risk Aversion and Sparsity Control |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2210.05108 |