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

This paper takes a uniform look at the customized applications of proximal point algorithm (PPA) to two classes of problems: the linearly constrained convex minimization problem with a generic or separable objective function and a saddle-point problem. We model these two classes of problems uniformly by a mixed variational inequality, and show how PPA with customized proximal parameters can yield favorable algorithms, which are able to exploit the structure of the models fully. Our customized PPA revisit turns out to be a uniform approach in designing a number of efficient algorithms, which are competitive with, or even more efficient than some benchmark methods in the existing literature such as the augmented Lagrangian method, the alternating direction method, the split inexact Uzawa method, and a class of primal-dual methods, etc. From the PPA perspective, the global convergence and the O(1/t) convergence rate for this series of algorithms are established in a uniform way.