Alternating Gradient-Type Algorithm for Bilevel Optimization with Inexact Lower-Level Solutions via Moreau Envelope-based Reformulation
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910209984692224 |
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| author | Bai, Xiaoning Zeng, Shangzhi Zhang, Jin Zhang, Lezhi |
| author_facet | Bai, Xiaoning Zeng, Shangzhi Zhang, Jin Zhang, Lezhi |
| contents | In this paper, we study a class of bilevel optimization problems where the lower-level problem is a convex composite optimization model, which arises in various applications, including bilevel hyperparameter selection for regularized regression models. To solve these problems, we propose an Alternating Gradient-type algorithm with Inexact Lower-level Solutions (AGILS) based on a Moreau envelope-based reformulation of the bilevel optimization problem. The proposed algorithm does not require exact solutions of the lower-level problem at each iteration, improving computational efficiency. We prove the convergence of AGILS to stationary points and, under the Kurdyka-Łojasiewicz (KL) property, establish its sequential convergence. Numerical experiments, including a toy example and a bilevel hyperparameter selection problem for the sparse group Lasso model, demonstrate the effectiveness of the proposed AGILS. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_18929 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Alternating Gradient-Type Algorithm for Bilevel Optimization with Inexact Lower-Level Solutions via Moreau Envelope-based Reformulation Bai, Xiaoning Zeng, Shangzhi Zhang, Jin Zhang, Lezhi Optimization and Control 90C26, 90C30, 49M37, 65K05 In this paper, we study a class of bilevel optimization problems where the lower-level problem is a convex composite optimization model, which arises in various applications, including bilevel hyperparameter selection for regularized regression models. To solve these problems, we propose an Alternating Gradient-type algorithm with Inexact Lower-level Solutions (AGILS) based on a Moreau envelope-based reformulation of the bilevel optimization problem. The proposed algorithm does not require exact solutions of the lower-level problem at each iteration, improving computational efficiency. We prove the convergence of AGILS to stationary points and, under the Kurdyka-Łojasiewicz (KL) property, establish its sequential convergence. Numerical experiments, including a toy example and a bilevel hyperparameter selection problem for the sparse group Lasso model, demonstrate the effectiveness of the proposed AGILS. |
| title | Alternating Gradient-Type Algorithm for Bilevel Optimization with Inexact Lower-Level Solutions via Moreau Envelope-based Reformulation |
| topic | Optimization and Control 90C26, 90C30, 49M37, 65K05 |
| url | https://arxiv.org/abs/2412.18929 |