Nonlinear Proximal Point Algorithms Using Bregman Functions, with Applications to Convex Programming

Abstract

A Bregman function is a strictly convex, differentiable function that induces a well-behaved distance measure or D-function on Euclidean space. This paper shows that, for every Bregman function, there exists a “nonlinear” version of the proximal point algorithm, and presents an accompanying convergence theory. Applying this generalization of the proximal point algorithm to convex programming, one obtains the D-function proximal minimization algorithm of Censor and Zenios, and a wide variety of new multiplier methods. These multiplier methods are different from those studied by Kort and Bertsekas, and include nonquadratic variations on the proximal method of multipliers.