On the Linear Convergence to Weak/Standard d-Stationary Points of DCA-Based Algorithms for Structured Nonsmooth DC Programming

Hongbo Dong () and Min Tao ()
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Hongbo Dong: Sunnyvale
Min Tao: Nanjing University

Journal of Optimization Theory and Applications, 2021, vol. 189, issue 1, No 9, 190-220

Abstract: Abstract We consider a class of structured nonsmooth difference-of-convex minimization. We allow nonsmoothness in both the convex and concave components in the objective function, with a finite max structure in the concave part. Our focus is on algorithms that compute a (weak or standard) d(irectional)-stationary point as advocated in a recent work of Pang et al. (Math Oper Res 42:95–118, 2017). Our linear convergence results are based on direct generalizations of the assumptions of error bounds and separation of isocost surfaces proposed in the seminal work of Luo and Tseng (Ann Oper Res 46–47:157–178, 1993), as well as one additional assumption of locally linear regularity regarding the intersection of certain stationary sets and dominance regions. An interesting by-product is to present a sharper characterization of the limit set of the basic algorithm proposed by Pang et al., which fits between d-stationarity and global optimality. We also discuss sufficient conditions under which these assumptions hold. Finally, we provide several realistic and nontrivial statistical learning models where all assumptions hold.

Keywords: Difference-of-convex programming; Nonsmooth; Difference-of-convex algorithm; Linear convergence; Error bound; 90C30; 90C26 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10957-021-01827-w

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