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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.

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