Approximate bregman proximal gradient algorithm for relatively smooth nonconvex optimization

Shota Takahashi () and Akiko Takeda ()
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Shota Takahashi: The University of Tokyo
Akiko Takeda: The University of Tokyo

Computational Optimization and Applications, 2025, vol. 90, issue 1, No 8, 227-256

Abstract: Abstract In this paper, we propose the approximate Bregman proximal gradient algorithm (ABPG) for solving composite nonconvex optimization problems. ABPG employs a new distance that approximates the Bregman distance, making the subproblem of ABPG simpler to solve compared to existing Bregman-type algorithms. The subproblem of ABPG is often expressed in a closed form. Similarly to existing Bregman-type algorithms, ABPG does not require the global Lipschitz continuity for the gradient of the smooth part. Instead, assuming the smooth adaptable property, we establish the global subsequential convergence under standard assumptions. Additionally, assuming that the Kurdyka–Łojasiewicz property holds, we prove the global convergence for a special case. Our numerical experiments on the $$\ell _p$$ ℓ p regularized least squares problem, the $$\ell _p$$ ℓ p loss problem, and the nonnegative linear system show that ABPG outperforms existing algorithms especially when the gradient of the smooth part is not globally Lipschitz or even locally Lipschitz continuous.

Keywords: Composite nonconvex nonsmooth optimization; Bregman proximal gradient algorithms; Kurdyka–Łojasiewicz property; $$\ell _p$$ ℓ p regularized least squares problem; $$\ell _p$$ ℓ p loss problem; nonnegative linear system; Kullback–Leibler divergence; 90C26; 49M37; 65K05 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10589-024-00618-z

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