A primal–dual prediction–correction algorithm for saddle point optimization

• Published in: Journal of global optimization Vol. 66; no. 3; pp. 573 - 583
• Main Authors: He, Hongjin, Desai, Jitamitra, Wang, Kai
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-11-2016; Springer; Springer Nature B.V
• Subjects: Algorithms; By products; Byproducts; Computational geometry; Computer Science; Convergence; Estimates; Euclidean space; Imaging; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Real Functions; Saddle points; Studies; Texts; Projection method; Convergence rate; Saddle point problem; Prediction–correction algorithm; Primal–dual algorithm
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-016-0437-1

In this paper, we introduce a new primal–dual prediction–correction algorithm for solving a saddle point optimization problem, which serves as a bridge between the algorithms proposed in Cai et al. (J Glob Optim 57:1419–1428,
2013
) and He and Yuan (SIAM J Imaging Sci 5:119–149,
2012
). An interesting byproduct of the proposed method is that we obtain an easily implementable projection-based primal–dual algorithm, when the primal and dual variables belong to simple convex sets. Moreover, we establish the worst-case
O
(
1
/
t
)
convergence rate result in an ergodic sense, where
t
represents the number of iterations.