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

Lei Wu ()
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Lei Wu: Jiangxi Normal University

Journal of Global Optimization, 2020, vol. 76, issue 1, No 7, 137-153

Abstract: 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.

Keywords: Structured optimization; Residual-based algorithm; Stationary point; Iteration-complexity; 90C26; 90C06; 90C30; 49M05 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10898-019-00776-z

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