By Mifflin R., Sagastizabal C.

For convex minimization we introduce an set of rules in line with VU-space decomposition. the strategy makes use of a package subroutine to generate a series of approximate proximal issues. while a primal-dual tune resulting in an answer and 0 subgradient pair exists, those issues approximate the primal music issues and provides the algorithm's V, or corrector, steps. The subroutine additionally approximates twin music issues which are U-gradients wanted for the method's U-Newton predictor steps. With the inclusion of an easy line seek the ensuing set of rules is proved to be globally convergent. The convergence is superlinear if the primal-dual music issues and the objective's U-Hessian are approximated good adequate.

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Then y - x _< l Y - x[ <_ 1 and hence if(y) + f " ' ( y ) ( y - x) - e x p ( - y ) ( 1 - (y - x)) > 0 > - e x p ( - y ) = f'"(y). (ii) imply that Df is separately convex. 5. (Fermi-Dirac entropy) Let f (x) - x ln(x) + (1 - x)ln(1 - x) on I - (0, 1). Then f"(x) - 1/(x(1 - x)). Thus 1/f"(x) = x(1 - x) is concave (but not affine). (i), D f is jointly convex. 6. 3 "barely" produces a jointly convex Bregman distance, by which we mean that the inequality (j) is always an equality. , there exist real a, ~ such that 1 f"(x) - a x + ~ > 0, for every x E I.

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