A smoothed proximal trust-region algorithm for nonconvex optimization problems with $L^p$-regularization, $p\in (0,1)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908497128456192 |
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| author | Antil, Harbir Lentz, Anna |
| author_facet | Antil, Harbir Lentz, Anna |
| contents | We investigate a trust-region algorithm to solve a nonconvex optimization problem with $L^p$-regularization for $p\in(0,1)$. The algorithm relies on descent properties of a so-called generalized Cauchy point that can be obtained efficiently by a line search along a suitable proximal path. To handle the nonconvexity and nonsmoothness of the $L^p$-pseudonorm, we replace it by a smooth approximation and construct a convex upper bound of that approximation. This enables us to use results of a trust-region method for composite problems with a convex nonsmooth term. We prove convergence properties of the resulting smoothed proximal trust-region algorithm and investigate its performance in some numerical examples. Furthermore, approximate subproblem solvers for the arising trust-region subproblems are considered. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_15446 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A smoothed proximal trust-region algorithm for nonconvex optimization problems with $L^p$-regularization, $p\in (0,1)$ Antil, Harbir Lentz, Anna Optimization and Control 49M15, 49M37, 90C26, 65K10, 65K05 We investigate a trust-region algorithm to solve a nonconvex optimization problem with $L^p$-regularization for $p\in(0,1)$. The algorithm relies on descent properties of a so-called generalized Cauchy point that can be obtained efficiently by a line search along a suitable proximal path. To handle the nonconvexity and nonsmoothness of the $L^p$-pseudonorm, we replace it by a smooth approximation and construct a convex upper bound of that approximation. This enables us to use results of a trust-region method for composite problems with a convex nonsmooth term. We prove convergence properties of the resulting smoothed proximal trust-region algorithm and investigate its performance in some numerical examples. Furthermore, approximate subproblem solvers for the arising trust-region subproblems are considered. |
| title | A smoothed proximal trust-region algorithm for nonconvex optimization problems with $L^p$-regularization, $p\in (0,1)$ |
| topic | Optimization and Control 49M15, 49M37, 90C26, 65K10, 65K05 |
| url | https://arxiv.org/abs/2508.15446 |