Computational Optimization and Applications

, Volume 59, Issue 1–2, pp 135–161

Customized proximal point algorithms for linearly constrained convex minimization and saddle-point problems: a unified approach

Article

DOI: 10.1007/s10589-013-9616-x

Cite this article as:
Gu, G., He, B. & Yuan, X. Comput Optim Appl (2014) 59: 135. doi:10.1007/s10589-013-9616-x

Abstract

This paper focuses on some customized applications of the proximal point algorithm (PPA) to two classes of problems: the convex minimization problem with linear constraints and a generic or separable objective function, and a saddle-point problem. We treat these two classes of problems uniformly by a mixed variational inequality, and show how the application of PPA with customized metric proximal parameters can yield favorable algorithms which are able to make use of the models’ structures effectively. Our customized PPA revisit turns out to unify some algorithms including some existing ones in the literature and some new ones to be proposed. From the PPA perspective, we establish the global convergence and a worst-case O(1/t) convergence rate for this series of algorithms in a unified way.

Keywords

Convex minimization Saddle-point problem Proximal point algorithm Convergence rate Customized algorithms Splitting algorithms 

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingChina
  2. 2.International Center of Management Science and Engineering and Department of MathematicsNanjing UniversityNanjingChina
  3. 3.Department of MathematicsHong Kong Baptist UniversityHong KongChina

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