A Residual-Based Algorithm for Solving a Class of Structured Nonsmooth Optimization Problems.

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.