Global convergence of model function based Bregman proximal minimization algorithms

Mahesh Chandra Mukkamala (), Jalal Fadili () and Peter Ochs ()
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Mahesh Chandra Mukkamala: University of Tübingen
Jalal Fadili: Normandie Univ, ENSICAEN, CNRS, GREYC
Peter Ochs: University of Tübingen

Journal of Global Optimization, 2022, vol. 83, issue 4, No 5, 753-781

Abstract: 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 like $$\vert x^4-1 \vert $$ | x 4 - 1 | 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.

Keywords: Composite minimization; Bregman proximal minimization algorithms; Model function framework; Bregman distance; Global convergence; Kurdyka–Łojasiewicz property (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10898-021-01114-y

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