New Bregman proximal type algorithms for solving DC optimization problems

Difference of Convex (DC) optimization problems have objective functions that are differences between two convex functions. Representative ways of solving these problems are the proximal DC algorithms, which require that the convex part of the objective function have L -smoothness. In this article,...

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Vydáno v:Computational optimization and applications Ročník 83; číslo 3; s. 893 - 931
Hlavní autoři: Takahashi, Shota, Fukuda, Mituhiro, Tanaka, Mirai
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.12.2022
Springer Nature B.V
Témata:
Algorithms
Convex and Discrete Geometry
Critical point
Management Science
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Phase retrieval
Smoothness
Statistics
Kurdyka-Łojasiewicz inequality
Nonconvex optimization
90C30
65K05
Nonsmooth optimization
90C26
Bregman proximal DC algorithms
Difference-of-convex optimization
Bregman distances
ISSN:0926-6003, 1573-2894
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Shrnutí:Difference of Convex (DC) optimization problems have objective functions that are differences between two convex functions. Representative ways of solving these problems are the proximal DC algorithms, which require that the convex part of the objective function have L -smoothness. In this article, we propose the Bregman Proximal DC Algorithm (BPDCA) for solving large-scale DC optimization problems that do not possess L -smoothness. Instead, it requires that the convex part of the objective function has the L -smooth adaptable property that is exploited in Bregman proximal gradient algorithms. In addition, we propose an accelerated version, the Bregman Proximal DC Algorithm with extrapolation (BPDCAe), with a new restart scheme. We show the global convergence of the iterates generated by BPDCA(e) to a limiting critical point under the assumption of the Kurdyka-Łojasiewicz property or subanalyticity of the objective function and other weaker conditions than those of the existing methods. We applied our algorithms to phase retrieval, which can be described both as a nonconvex optimization problem and as a DC optimization problem. Numerical experiments showed that BPDCAe outperformed existing Bregman proximal-type algorithms because the DC formulation allows for larger admissible step sizes.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-022-00411-w