Forward-backward splitting under the light of generalized convexity
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909549674364928 |
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| author | Oikonomidis, Konstantinos Laude, Emanuel Patrinos, Panagiotis |
| author_facet | Oikonomidis, Konstantinos Laude, Emanuel Patrinos, Panagiotis |
| contents | In this paper we present a unifying framework for continuous optimization methods grounded in the concept of generalized convexity. Utilizing the powerful theory of $Φ$-convexity, we propose a conceptual algorithm that extends the classical difference-of-convex method, encompassing a broad spectrum of optimization algorithms. Relying exclusively on the tools of generalized convexity we develop a gap function analysis that strictly characterizes the decrease of the function values, leading to simplified and unified convergence results. As an outcome of this analysis, we naturally obtain a generalized PL inequality which ensures $q$-linear convergence rates of the proposed method, incorporating various well-established conditions from the existing literature. Moreover we propose a $Φ$-Bregman proximal point interpretation of the scheme that allows us to capture conditions that lead to sublinear rates under convexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18098 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Forward-backward splitting under the light of generalized convexity Oikonomidis, Konstantinos Laude, Emanuel Patrinos, Panagiotis Optimization and Control In this paper we present a unifying framework for continuous optimization methods grounded in the concept of generalized convexity. Utilizing the powerful theory of $Φ$-convexity, we propose a conceptual algorithm that extends the classical difference-of-convex method, encompassing a broad spectrum of optimization algorithms. Relying exclusively on the tools of generalized convexity we develop a gap function analysis that strictly characterizes the decrease of the function values, leading to simplified and unified convergence results. As an outcome of this analysis, we naturally obtain a generalized PL inequality which ensures $q$-linear convergence rates of the proposed method, incorporating various well-established conditions from the existing literature. Moreover we propose a $Φ$-Bregman proximal point interpretation of the scheme that allows us to capture conditions that lead to sublinear rates under convexity. |
| title | Forward-backward splitting under the light of generalized convexity |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2503.18098 |