Computation of second-order directional stationary points for group sparse optimization. (3rd March 2020)

• Record Type: Journal Article
• Authors: Peng, Dingtao; Chen, Xiaojun
• Is Part Of: Optimization methods and software. Volume 35:Number 2(2020)
• Page Start: 348
• Page End: 376
• Publication Date: 2020-03-03
• Subjects: Group sparse optimization -- nonconvex and nonsmooth optimization -- composite folded concave penalty -- directional stationary point -- smoothing method; 90C26 -- 90C46; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2019.1684492
• Languages: English
• ISSNs: 1055-6788

Abstract

ABSTRACT: We consider a nonconvex and nonsmooth group sparse optimization problem where the penalty function is the sum of compositions of a folded concave function and the ℓ 2 vector norm for each group variable. We show that under some mild conditions a first-order directional stationary point is a strict local minimizer that fulfils the first-order growth condition, and a second-order directional stationary point is a strong local minimizer that fulfils the second-order growth condition. In order to compute second-order directional stationary points, we construct a twice continuously differentiable smoothing problem and show that any accumulation point of the sequence of second-order stationary points of the smoothing problem is a second-order directional stationary point of the original problem. We give numerical examples to illustrate how to compute a second-order directional stationary point by the smoothing method.