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Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming. (English) Zbl 0807.47036
Summary: 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.

##### MSC:
 47H05 Monotone operators and generalizations 49M37 Numerical methods based on nonlinear programming 90C25 Convex programming
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