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Accelerated inexact composite gradient methods for nonconvex spectral optimization problems

Author

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  • Weiwei Kong

    (Oak Ridge National Laboratory)

  • Renato D. C. Monteiro

    (Georgia Institute of Technology)

Abstract

This paper presents two inexact composite gradient methods, one inner accelerated and another doubly accelerated, for solving a class of nonconvex spectral composite optimization problems. More specifically, the objective function for these problems is of the form $$f_{1}+f_{2}+h$$ f 1 + f 2 + h , where $$f_{1}$$ f 1 and $$f_{2}$$ f 2 are differentiable nonconvex matrix functions with Lipschitz continuous gradients, $$h$$ h is a proper closed convex matrix function, and both $$f_{2}$$ f 2 and $$h$$ h can be expressed as functions that operate on the singular values of their inputs. The methods essentially use an accelerated composite gradient method to solve a sequence of proximal subproblems involving the linear approximation of $$f_{1}$$ f 1 and the singular value functions underlying $$f_{2}$$ f 2 and $$h$$ h . Unlike other composite gradient-based methods, the proposed methods take advantage of both the composite and spectral structure underlying the objective function in order to efficiently generate their solutions. Numerical experiments are presented to demonstrate the practicality of these methods on a set of real-world and randomly generated spectral optimization problems.

Suggested Citation

  • Weiwei Kong & Renato D. C. Monteiro, 2022. "Accelerated inexact composite gradient methods for nonconvex spectral optimization problems," Computational Optimization and Applications, Springer, vol. 82(3), pages 673-715, July.
  • Handle: RePEc:spr:coopap:v:82:y:2022:i:3:d:10.1007_s10589-022-00377-9
    DOI: 10.1007/s10589-022-00377-9
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    References listed on IDEAS

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    1. Renato D. C. Monteiro & Camilo Ortiz & Benar F. Svaiter, 2016. "An adaptive accelerated first-order method for convex optimization," Computational Optimization and Applications, Springer, vol. 64(1), pages 31-73, May.
    2. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Bo Wen & Xiaojun Chen & Ting Kei Pong, 2018. "A proximal difference-of-convex algorithm with extrapolation," Computational Optimization and Applications, Springer, vol. 69(2), pages 297-324, March.
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    Cited by:

    1. Jiaming Liang & Renato D. C. Monteiro, 2023. "Average curvature FISTA for nonconvex smooth composite optimization problems," Computational Optimization and Applications, Springer, vol. 86(1), pages 275-302, September.

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