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More than first-order developments of convex functions: primal-dual relations. (English) Zbl 0868.49014
Summary: The subject of this paper concerns the remainder term in the first-order development of a (finite-valued) convex function. We study functions for which this term is comparable to a squared norm and we relate it to the corresponding remainder term of the conjugate function. We show that a convex function satisfies a quadratic growth condition if and only if its subdifferential satisfies a linear growth condition. Finally, we define a new concept of “tangential regularization”, involving a local decomposition of ${ℝ}^{N}$, along the subspace where the function is “smooth” and the subspace parallel to the subdifferential.
##### MSC:
 49J52 Nonsmooth analysis (other weak concepts of optimality) 26B25 Convexity and generalizations (several real variables) 52A41 Convex functions and convex programs (convex geometry)