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Global convergence of model function based Bregman proximal minimization algorithms

Author

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  • Mahesh Chandra Mukkamala

    (University of Tübingen)

  • Jalal Fadili

    (Normandie Univ, ENSICAEN, CNRS, GREYC)

  • Peter Ochs

    (University of Tübingen)

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.

Suggested Citation

  • Mahesh Chandra Mukkamala & Jalal Fadili & Peter Ochs, 2022. "Global convergence of model function based Bregman proximal minimization algorithms," Journal of Global Optimization, Springer, vol. 83(4), pages 753-781, August.
    • Handle: RePEc:spr:jglopt:v:83:y:2022:i:4:d:10.1007_s10898-021-01114-y
    DOI: 10.1007/s10898-021-01114-y
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    References listed on IDEAS

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    1. Haihao Lu & Robert M. Freund & Yurii Nesterov, 2018. "Relatively smooth convex optimization by first-order methods, and applications," LIDAM Reprints CORE 2965, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Peter Ochs & Jalal Fadili & Thomas Brox, 2019. "Non-smooth Non-convex Bregman Minimization: Unification and New Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 244-278, April.
    3. Yurii Nesterov, 2018. "Smooth Convex Optimization," Springer Optimization and Its Applications, in: Lectures on Convex Optimization, edition 2, chapter 0, pages 59-137, Springer.
    4. NESTEROV, Yurii, 2007. "Gauss-Newton scheme with worst case guarantees for global performance," LIDAM Reprints CORE 1952, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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