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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. Yurii Nesterov, 2018. "Smooth Convex Optimization," Springer Optimization and Its Applications, in: Lectures on Convex Optimization, edition 2, chapter 0, pages 59-137, Springer.
    3. 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).
    4. 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.
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