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<p>We investigate the convergence properties of exact and inexact forward–backward algorithms to minimize the sum of two weakly convex functions defined on a Hilbert space, where one has a Lipschitz-continuous gradient. We show that the exact forward–backward algorithm locally converges strongly to a global solution, provided that the objective function satisfies a sharpness condition. For the inexact forward–backward algorithm, the same condition ensures that the distance from the iterates to the solution set approaches a positive threshold depending on the accuracy level of the proximal computations. As an application of the considered setting, we provide numerical experiments related to discrete tomography.</p>

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Forward–Backward Algorithms for Weakly Convex Problems

  • 期刊论文

  • 发表日期：

    2026年4月17日

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  • Ewa Bednarczuk,
  • Giovanni Bruccola,
  • Gabriele Scrivanti,
  • The Hung Tran

摘要

We investigate the convergence properties of exact and inexact forward–backward algorithms to minimize the sum of two weakly convex functions defined on a Hilbert space, where one has a Lipschitz-continuous gradient. We show that the exact forward–backward algorithm locally converges strongly to a global solution, provided that the objective function satisfies a sharpness condition. For the inexact forward–backward algorithm, the same condition ensures that the distance from the iterates to the solution set approaches a positive threshold depending on the accuracy level of the proximal computations. As an application of the considered setting, we provide numerical experiments related to discrete tomography.