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

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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 \(h(x)=0\), where h is continuous but not differentiable, and moreover \(-h(x)\) is a descent direction of the objective at \(x\in \mathbb {R}^n\) if \(h(x)\ne 0\). Then by using \(-h(x)\) 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 \(\epsilon \) first-order stationary point is \(O(\epsilon ^{-2})\). Numerical results have shown the efficiency of this algorithm.

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Acknowledgements

The authors would like to thank the anonymous referee and the editor for their valuable suggestions and comments.

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Correspondence to Lei Wu.

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The work was supported by the National Nature Science Foundation of People’s Republic of China (11501265 and 11761037), the Nature Science Foundation of Jiangxi (No. 20161BAB211011), and the Foundation of Department of Education Jiangxi Province (No. GJJ150314)

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Wu, L. A residual-based algorithm for solving a class of structured nonsmooth optimization problems. J Glob Optim 76, 137–153 (2020). https://doi.org/10.1007/s10898-019-00776-z

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  • DOI: https://doi.org/10.1007/s10898-019-00776-z

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