Please use this identifier to cite or link to this item: https://hdl.handle.net/10419/287413 
Year of Publication: 
2021
Citation: 
[Journal:] Journal of Global Optimization [ISSN:] 1573-2916 [Volume:] 83 [Issue:] 4 [Publisher:] Springer US [Place:] New York, NY [Year:] 2021 [Pages:] 753-781
Publisher: 
Springer US, New York, NY
Abstract: 
Lipschitz continuity of the gradient mapping of a continuously differentiable function plays a crucial role in designing various optimization algorithms. However, many functions arising in practical applications such as low rank matrix factorization or deep neural network problems do not have a Lipschitz continuous gradient. This led to the development of a generalized notion known as the L-smad property, which is based on generalized proximity measures called Bregman distances. However, the L-smad property cannot handle nonsmooth functions, for example, simple nonsmooth functions like4-1 ocumentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert x -1 \vert $$\end{document} and also many practical composite problems are out of scope. We fix this issue by proposing the MAP property, which generalizes the L-smad property and is also valid for a large class of structured nonconvex nonsmooth composite problems. Based on the proposed MAP property, we propose a globally convergent algorithm called Model BPG, that unifies several existing algorithms. The convergence analysis is based on a new Lyapunov function. We also numerically illustrate the superior performance of Model BPG on standard phase retrieval problems and Poisson linear inverse problems, when compared to a state of the art optimization method that is valid for generic nonconvex nonsmooth optimization problems.
Subjects: 
Composite minimization
Bregman proximal minimization algorithms
Model function framework
Bregman distance
Global convergence
Kurdyka–Łojasiewicz property
Persistent Identifier of the first edition: 
Creative Commons License: 
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Document Type: 
Article
Document Version: 
Published Version

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