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A residual-based algorithm for solving a class of structured nonsmooth optimization problems

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  • Lei Wu

    (Jiangxi Normal University)

Abstract

In this paper, we consider a class of structured nonsmooth optimization problem in which the first component of the objective is a smooth function while the second component is the sum of one-dimensional nonsmooth functions. We first verify that every minimizer of this problem is a solution of an equation $$h(x)=0$$h(x)=0, where h is continuous but not differentiable, and moreover $$-h(x)$$-h(x) is a descent direction of the objective at $$x\in \mathbb {R}^n$$x∈Rn if $$h(x)\ne 0$$h(x)≠0. Then by using $$-h(x)$$-h(x) as a search direction, we propose a residual-based algorithm for solving this problem. Under proper conditions, we verify that any accumulation point of the sequence of iterates generated by our algorithm is a first-order stationary point of the problem. Additionally, we prove that the worst-case iteration-complexity for finding an $$\epsilon $$ϵ first-order stationary point is $$O(\epsilon ^{-2})$$O(ϵ-2). Numerical results have shown the efficiency of this algorithm.

Suggested Citation

  • Lei Wu, 2020. "A residual-based algorithm for solving a class of structured nonsmooth optimization problems," Journal of Global Optimization, Springer, vol. 76(1), pages 137-153, January.
    • Handle: RePEc:spr:jglopt:v:76:y:2020:i:1:d:10.1007_s10898-019-00776-z
    DOI: 10.1007/s10898-019-00776-z
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    References listed on IDEAS

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    1. Xiaojun Chen & Weijun Zhou, 2014. "Convergence of the reweighted ℓ 1 minimization algorithm for ℓ 2 –ℓ p minimization," Computational Optimization and Applications, Springer, vol. 59(1), pages 47-61, October.
    2. Zhaosong Lu & Xiaorui Li, 2018. "Sparse Recovery via Partial Regularization: Models, Theory, and Algorithms," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1290-1316, November.
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