Convergence of alternating direction method for minimizing sum of two nonconvex functions with linear constraints

ABSTRACT

The efficiency of the classic alternating direction method of multipliers has been exhibited by various applications for large-scale separable optimization problems, both for convex objective functions and for nonconvex objective functions. While there are a lot of convergence analysis for the convex case, the nonconvex case is still an open problem and the research for this case is in its infancy. In this paper, we give a partial answer on this problem. Specially, under the assumption that the associated function satisfies the Kurdyka–Łojasiewicz inequality, we prove that the iterative sequence generated by the alternating direction method converges to a critical point of the problem, provided that the penalty parameter is greater than 2L, where L is the Lipschitz constant of the gradient of one of the involved functions. Under some further conditions on the problem's data, we also analyse the convergence rate of the algorithm.

KEYWORDS:

• Alternating direction method of multipliers
• Kurdyka–Łojasiewicz inequality
• nonconvex optimization
• linear constraints
• global convergence

2010 AMS SUBJECT CLASSIFICATIONS:

• 90C26
• 65K10
• 49J52
• 49M27

Acknowledgments

The first author was supported by the Natural Science Foundation of China [Grant No. 11371015] and the Graduate Student Research and Innovation Program of Jiangsu Province of China [Grant No. KYLX-069]. The second author was supported by a project funded by PAPD of Jiangsu Higher Education Institutions and the Natural Science Foundation of China [Grant No. 11371197, 11431002]. The third author was supported by the Natural Science Foundation of China [Grant No. 11501301], Jiangsu Planned Projects for Postdoctoral Research Funds [Grant No. 1501071B], and the Foundation of Jiangsu Key Lab for NSLSCS [Grant No. 201601].

Disclosure statement

No potential conflict of interest was reported by the authors.