A generalized alternating direction method of multipliers with semi-proximal terms for convex composite conic programming
- 336 Downloads
In this paper, we propose a generalized alternating direction method of multipliers (ADMM) with semi-proximal terms for solving a class of convex composite conic optimization problems, of which some are high-dimensional, to moderate accuracy. Our primary motivation is that this method, together with properly chosen semi-proximal terms, such as those generated by the recent advance of block symmetric Gauss–Seidel technique, is capable of tackling these problems. Moreover, the proposed method, which relaxes both the primal and the dual variables in a natural way with a common relaxation factor in the interval of (0, 2), has the potential of enhancing the performance of the classic ADMM. Extensive numerical experiments on various doubly non-negative semidefinite programming problems, with or without inequality constraints, are conducted. The corresponding results showed that all these multi-block problems can be successively solved, and the advantage of using the relaxation step is apparent.
KeywordsConvex composite conic programming Alternating direction method of multipliers Doubly non-negative semidefinite programming Relaxation Semi-proximal terms
Mathematics Subject Classification90C22 90C25 90C06 65K05
We would like to thank the anonymous referees and the associate editor for their useful comments and suggestions which improved this paper greatly. We are very grateful to Professor Defeng Sun at the Hong Kong Polytechnic University for sharing his knowledge with us on topics covered in this paper and beyond. The research of Y. Xiao and L. Chen was supported by the China Scholarship Council while they were visiting the National University of Singapore. The research of Y. Xiao was supported by the Major State Basic Research Development Program of China (973 Program) (Grant No. 2015CB856003), and the National Natural Science Foundation of China (Grant No. 11471101). The research of L. Chen was supported by the Fundamental Research Funds for Central Universities and the National Natural Science Foundation of China (Grant No. 11271117). The research of D. Li was supported by the National Natural Science Foundation of China (Grant No. 11371154 and 11771157).
- 1.Chen, C.H.: Numerical algorithms for a class of matrix norm approximation problems. Ph.D. Thesis, Department of Mathematics, Nanjing University, Nanjing, China. http://www.math.nus.edu.sg/~matsundf/Thesis_Caihua.pdf (2012)
- 11.Gabay, D.: Studies in mathematics and its applications. In: Fortin, M., Glowinski, R. (eds.) Applications of the method of multipliers to variational inequalities in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, vol. 15, pp. 299–331. Elsevier, Amsterdam (1983)Google Scholar
- 13.Glowinski, R.: Lectures on numerical methods for non-linear variational problems. Published for the Tata Institute of Fundamental Research, Bombay [by] Springer (1980)Google Scholar
- 14.Glowinski, R. and Marroco, A.: Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problèmes de Dirichlet non linéaires. Revue française d’atomatique, Informatique Recherche Opérationelle. Analyse Numérique, 9(2), 41–76 (1975)Google Scholar
- 18.Hong, M., Chang, T.-H., Wang, X., Razaviyayn, M., Ma, S. and Luo, Z.-Q.: A block successive upper bound minimization method of multipliers for linearly constrained convex optimization. arXiv:1401.7079 (2014)
- 19.Li, X.D., Sun, D.F., Toh. K.-C.: QSDPNAL: A two-phase Newton-CG proximal augmented Lagrangian method for convex quadratic semidefinite programming problems, arXiv:1512.08872 (2015)
- 20.Li, X.D.: A two-phase augmented Lagrangian method for convex composite quadratic programming, PhD Thesis, Department of Mathematics, National University of Singapore (2015)Google Scholar
- 28.Powell, M.J.D.: Optimization. In: Fletcher, R. (ed.) A method for nonlinear constraints in minimization problems, pp. 283–298. Academic Press, London (1969)Google Scholar
- 32.Rockafellar, R.T.: Monotone operators and augmented lagrangian methods in nonlinear programming. In: Mangasarian, O.L., Meyer, R.M., Robinson, S.M. (eds.) Nonlinear Programming 3, pp. 1–25. Academic Press, New York (1977)Google Scholar