Proximal variable smoothing method for three-composite nonconvex nonsmooth minimization with a linear operator

In this paper, we consider a class of three-composite nonconvex nonsmooth optimization problems, where one of nonsmooth functions is further composed with linear operator. Based on the variable smoothing method, as well as first-order methods with suitable majorization techniques, we propose a proxi...

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Vydáno v:Numerical algorithms Ročník 96; číslo 1; s. 237 - 266
Hlavní autoři: Liu, Yuncheng, Xia, Fuquan
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.05.2024
Springer Nature B.V
Témata:
Algebra
Algorithms
Approximation
Computer Science
Convex analysis
Linear operators
Mathematical analysis
Mathematical functions
Methods
Numeric Computing
Numerical Analysis
Operators (mathematics)
Original Paper
Smoothing
Theory of Computation
Proximal mapping
Variable smoothing
Nonconvex nonsmooth optimization
Incremental aggregated gradient
ISSN:1017-1398, 1572-9265
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Shrnutí:In this paper, we consider a class of three-composite nonconvex nonsmooth optimization problems, where one of nonsmooth functions is further composed with linear operator. Based on the variable smoothing method, as well as first-order methods with suitable majorization techniques, we propose a proximal variable smoothing gradient (ProxVSG) method for solving this kind of problem. The ProxVSG can be implemented efficiently, thanks to the fact that at each iteration, one just separately computes the proximal mapping of each nonsmooth function, rather than that of the sum of these nonsmooth functions. Furthermore, within our broad and flexible analysis framework, we propose a new proximal variable smoothing incremental aggregated gradient (ProxVSIAG) generalizations of the ProxVSG. In ProxVSIAG, an incremental aggregated estimate of the gradient is used, instead of the full gradient. Under suitable assumptions, we prove a complexity of O ( ϵ - 3 ) to achieve an ϵ -approximate solution. Preliminary numerical experiments show the efficiency of the proposed method.
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-023-01645-3