Customized Proximal Point Algorithms for Convex Optimization

The proximal point algorithm (PPA) plays a fundamental theoretical and algorithmic role in Optimization. This project revisits the PPA to develop efficient PPA-based algorithms for some basic convex optimization models. Our philosophy of algorithmic design is to customize the proximal parameters (in metric form) in accordance with the model structures, and thus convert the classical PPA in abstract form into some customized algorithms in concrete form. Further, some advances in the PPA literature will be exploited to accelerate these customized PPAs. The splitting nature of these new algorithms makes it possible to take full advantage of the model structure, to yield easy subproblems with closed-form solutions.

Numerically, we expect these customized algorithms to compete with or be even faster than some benchmark methods. The efficiency of these customized algorithms will be verified by application to problems arising in various areas. Theoretically, we will analyze their global convergence and estimate their rate of convergence.