Higher-order tensor methods for minimizing difference of convex functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916086868344832 |
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| author | Necoara, Ion |
| author_facet | Necoara, Ion |
| contents | Higher-order tensor methods were recently proposed for minimizing smooth convex and nonconvex functions. Higher-order algorithms accelerate the convergence of the classical first-order methods thanks to the higher-order derivatives used in the updates. The purpose of this paper is twofold. Firstly, to show that the higher-order algorithmic framework can be generalized and successfully applied to (nonsmooth) difference of convex functions, namely, those that can be expressed as the difference of two smooth convex functions and a possibly nonsmooth convex one. We also provide examples when the subproblem can be solved efficiently, even globally. Secondly, to derive a complete convergence analysis for our higher-order difference of convex functions (HO-DC) algorithm. In particular, we prove that any limit point of the HO-DC iterative sequence is a critical point of the problem under consideration, the corresponding objective value is monotonically decreasing and the minimum value of the norms of its subgradients converges globally to zero at a sublinear rate. The sublinear or linear convergence rates of the iterations are obtained under the Kurdyka-Lojasiewicz property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05063 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher-order tensor methods for minimizing difference of convex functions Necoara, Ion Optimization and Control Higher-order tensor methods were recently proposed for minimizing smooth convex and nonconvex functions. Higher-order algorithms accelerate the convergence of the classical first-order methods thanks to the higher-order derivatives used in the updates. The purpose of this paper is twofold. Firstly, to show that the higher-order algorithmic framework can be generalized and successfully applied to (nonsmooth) difference of convex functions, namely, those that can be expressed as the difference of two smooth convex functions and a possibly nonsmooth convex one. We also provide examples when the subproblem can be solved efficiently, even globally. Secondly, to derive a complete convergence analysis for our higher-order difference of convex functions (HO-DC) algorithm. In particular, we prove that any limit point of the HO-DC iterative sequence is a critical point of the problem under consideration, the corresponding objective value is monotonically decreasing and the minimum value of the norms of its subgradients converges globally to zero at a sublinear rate. The sublinear or linear convergence rates of the iterations are obtained under the Kurdyka-Lojasiewicz property. |
| title | Higher-order tensor methods for minimizing difference of convex functions |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2401.05063 |