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Journal of Convex Analysis 09 (2002), No. 2, 563--579
Copyright Heldermann Verlag 2002

Proximal Points are on the Fast Track

Robert Mifflin
Dept. of Pure and Applied Mathematics, Washington State University, Pullman, WA 99164-3113, U.S.A.

Claudia Sagastizábal
National Institute of Pure and Applied Mathematics, Estada Dona Castorina 110, Jardim Botanico, Rio de Janeiro, RJ 22460-320, Brazil

For a convex function, we consider a space decomposition that allows us to identify a subspace on which a Lagrangian related to the function appears to be smooth. We study a particular trajectory, that we call a fast track, on which a certain second-order expansion of the function can be obtained. We show how to obtain such fast tracks for a general class of convex functions having primal-dual gradient structure. Finally, we show that for a point near a minimizer its corresponding proximal point is on the fast track.

Keywords: Convex minimization, proximal points, second-order derivatives, VU-decomposition.

MSC: 49K35, 49M27; 65K10, 90C25

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