An interior proximal gradient method for nonconvex optimization
We consider structured minimization problems subject to smooth inequality constraints and present a flexible algorithm that combines interior point (IP) and proximal gradient schemes. While traditional IP methods cannot cope with nonsmooth objective functions and proximal algorithms cannot handle co...
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| Format: | Article |
| Language: | English |
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Université de Montpellier
2024-07-01
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| Series: | Open Journal of Mathematical Optimization |
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| Online Access: | https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.30/ |
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| author | De Marchi, Alberto Themelis, Andreas |
| author_facet | De Marchi, Alberto Themelis, Andreas |
| author_sort | De Marchi, Alberto |
| collection | DOAJ |
| description | We consider structured minimization problems subject to smooth inequality constraints and present a flexible algorithm that combines interior point (IP) and proximal gradient schemes. While traditional IP methods cannot cope with nonsmooth objective functions and proximal algorithms cannot handle complicated constraints, their combined usage is shown to successfully compensate the respective shortcomings. We provide a theoretical characterization of the algorithm and its asymptotic properties, deriving convergence results for fully nonconvex problems, thus bridging the gap with previous works that successfully addressed the convex case. Our interior proximal gradient algorithm benefits from warm starting, generates strictly feasible iterates with decreasing objective value, and returns after finitely many iterations a primal-dual pair approximately satisfying suitable optimality conditions. As a byproduct of our analysis of proximal gradient iterations we demonstrate that a slight refinement of traditional backtracking techniques waives the need for upper bounding the stepsize sequence, as required in existing results for the nonconvex setting. |
| format | Article |
| id | doaj-art-99c9d7b6bc32414d957074d1e816cfe4 |
| institution | Kabale University |
| issn | 2777-5860 |
| language | English |
| publishDate | 2024-07-01 |
| publisher | Université de Montpellier |
| record_format | Article |
| series | Open Journal of Mathematical Optimization |
| spelling | doaj-art-99c9d7b6bc32414d957074d1e816cfe42025-02-07T14:01:17ZengUniversité de MontpellierOpen Journal of Mathematical Optimization2777-58602024-07-01512210.5802/ojmo.3010.5802/ojmo.30An interior proximal gradient method for nonconvex optimizationDe Marchi, Alberto0Themelis, Andreas1University of the Bundeswehr Munich Department of Aerospace Engineering, Institute of Applied Mathematics and Scientific Computing Werner-Heisenberg-Weg 39, 85577 Neubiberg, GermanyKyushu University Faculty of Information Science and Electrical Engineering (ISEE) 744 Motooka, Nishi-ku, 819-0395 Fukuoka, JapanWe consider structured minimization problems subject to smooth inequality constraints and present a flexible algorithm that combines interior point (IP) and proximal gradient schemes. While traditional IP methods cannot cope with nonsmooth objective functions and proximal algorithms cannot handle complicated constraints, their combined usage is shown to successfully compensate the respective shortcomings. We provide a theoretical characterization of the algorithm and its asymptotic properties, deriving convergence results for fully nonconvex problems, thus bridging the gap with previous works that successfully addressed the convex case. Our interior proximal gradient algorithm benefits from warm starting, generates strictly feasible iterates with decreasing objective value, and returns after finitely many iterations a primal-dual pair approximately satisfying suitable optimality conditions. As a byproduct of our analysis of proximal gradient iterations we demonstrate that a slight refinement of traditional backtracking techniques waives the need for upper bounding the stepsize sequence, as required in existing results for the nonconvex setting.https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.30/Nonsmooth nonconvex optimizationinterior point methodsproximal algorithmslocally Lipschitz gradient |
| spellingShingle | De Marchi, Alberto Themelis, Andreas An interior proximal gradient method for nonconvex optimization Open Journal of Mathematical Optimization Nonsmooth nonconvex optimization interior point methods proximal algorithms locally Lipschitz gradient |
| title | An interior proximal gradient method for nonconvex optimization |
| title_full | An interior proximal gradient method for nonconvex optimization |
| title_fullStr | An interior proximal gradient method for nonconvex optimization |
| title_full_unstemmed | An interior proximal gradient method for nonconvex optimization |
| title_short | An interior proximal gradient method for nonconvex optimization |
| title_sort | interior proximal gradient method for nonconvex optimization |
| topic | Nonsmooth nonconvex optimization interior point methods proximal algorithms locally Lipschitz gradient |
| url | https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.30/ |
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