An effective subgradient algorithm via Mifflin's line search for nonsmooth nonconvex multiobjective optimization
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arXiv
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| Natura: | Preprint |
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2024
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| author | Maleknia, Morteza Soleimani-damaneh, Majid |
| author_facet | Maleknia, Morteza Soleimani-damaneh, Majid |
| contents | We propose a descent subgradient algorithm for unconstrained nonsmooth nonconvex multiobjective optimization problems. To find a descent direction, we present an iterative process that efficiently approximates the Goldstein subdifferential of each objective function. To this end, we develop a new variant of Mifflin's line search in which the subgradients are arbitrary and its finite convergence is proved under a semismooth assumption. To reduce the number of subgradient evaluations, we employ a backtracking line search that identifies the objectives requiring an improvement in the current approximation of the Goldstein subdifferential. Meanwhile, for the remaining objectives, new subgradients are not computed. Unlike bundle-type methods, the proposed approach can handle nonconvexity without the need for algorithmic adjustments. Moreover, the quadratic subproblems have a simple structure, and hence the method is easy to implement. We analyze the global convergence of the proposed method and prove that any accumulation point of the generated sequence satisfies a necessary Pareto optimality condition. Furthermore, our convergence analysis addresses a theoretical challenge in a recently developed subgradient method. Through numerical experiments, we observe the practical capability of the proposed method and evaluate its efficiency when applied to a diverse range of nonsmooth test problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14905 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An effective subgradient algorithm via Mifflin's line search for nonsmooth nonconvex multiobjective optimization Maleknia, Morteza Soleimani-damaneh, Majid Optimization and Control We propose a descent subgradient algorithm for unconstrained nonsmooth nonconvex multiobjective optimization problems. To find a descent direction, we present an iterative process that efficiently approximates the Goldstein subdifferential of each objective function. To this end, we develop a new variant of Mifflin's line search in which the subgradients are arbitrary and its finite convergence is proved under a semismooth assumption. To reduce the number of subgradient evaluations, we employ a backtracking line search that identifies the objectives requiring an improvement in the current approximation of the Goldstein subdifferential. Meanwhile, for the remaining objectives, new subgradients are not computed. Unlike bundle-type methods, the proposed approach can handle nonconvexity without the need for algorithmic adjustments. Moreover, the quadratic subproblems have a simple structure, and hence the method is easy to implement. We analyze the global convergence of the proposed method and prove that any accumulation point of the generated sequence satisfies a necessary Pareto optimality condition. Furthermore, our convergence analysis addresses a theoretical challenge in a recently developed subgradient method. Through numerical experiments, we observe the practical capability of the proposed method and evaluate its efficiency when applied to a diverse range of nonsmooth test problems. |
| title | An effective subgradient algorithm via Mifflin's line search for nonsmooth nonconvex multiobjective optimization |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2406.14905 |