An Inertial Algorithm for DC Programming

• Published in: Set-valued and variational analysis Vol. 27; no. 4; pp. 895 - 919
• Main Authors: de Oliveira, Welington, Tcheou, Michel P.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.12.2019; Springer Nature B.V; Springer
• Subjects: Algorithms; Analysis; Convergence; Critical point; Mathematics; Mathematics and Statistics; Noise reduction; Optimization; Optimization and Control; Variational analysis; 49K99; 49J53; 49J52; DC programming; Nonsmooth optimization; 90C26
• ISSN: 1877-0533; 1877-0541
• DOI: 10.1007/s11228-018-0497-0

We consider nonsmooth optimization problems whose objective function is defined by the Difference of Convex (DC) functions. With the aim of computing critical points that are also d (irectional)-stationary for such a class of nonconvex programs we propose an algorithmic scheme equipped with an inertial-force procedure. In contrast to the classical DC algorithm of P. D. Tao and L. T. H. An, the proposed inertial DC algorithm defines trial points whose sequence of functional values is not necessary monotonically decreasing, a property that proves useful to prevent the algorithm from converging to a critical point that is not d -stationary. Moreover, our method can handle inexactness in the solution of convex subproblems yielding trial points. This is another property of practical interest that substantially reduces the computational burden to compute d -stationary/critical points of DC programs. Convergence analysis of the proposed algorithm yields global convergence to critical points, and convergence rate is established for the considered class of problems. Numerical experiments on large-scale (nonconvex and nonsmooth) image denoising models show that the proposed algorithm outperforms the classic one in this particular application, specifically in the case of piecewise constant images with neat edges such as QR codes.