Weaker conditions for subdifferential calculus of convex functions

Abstract

In this paper we establish new rules for the calculus of the subdifferential mapping of the sum of two convex functions. Our results are established under conditions which are at an intermediate level of generality among those leading to the Hiriart-Urruty and Phelps formula (Hiriart-Urruty and Phelps, 1993 [15]), involving the approximate subdifferential, and the stronger assumption used in the well-known Moreau-Rockafellar formula (Rockafellar 1970, [23]; Moreau 1966, [20]), which only uses the exact subdifferential. We give an application to derive asymptotic optimality conditions for convex optimization.CONICYT 1151003 1150909 Math-Amsud program 13MATH-01 2013 MINECO of Spain FEDER of EU MTM2014-59179-C2-1-P MTM2011-29064-C03-02 Australian Research Council DP16010085

Towards Supremum-Sum Subdifferential Calculus Free of Qualification Conditions

“…We close this paper with a refinement of the classical result due to Brøndsted (see (5)). The proof of the following corollary combines Theorem 4 and the arguments used in [8,Corollary 12].…”

“…Proposition 1 (see [5,Theorem 12]). Let f and g be two proper convex functions defined on X and satisfying f + g =f + g, together with…”

“…Assume now that g ≡ 0 and R + (dom f − x) is closed. Then, again using (7) Observe that the second part of the previous corollary applies when f is a polyhedral function, since in this case the set R + (dom f − x) is closed [5]. A characterization of this closedness property can be found in [14].…”

“…Related closedness conditions will be crucial in this paper, and they are addressed to relax the lsc property. For instance, it has been shown in [5] (see Propositions 1 and 2 below) that for every …”

Characterizations of the subdifferential of convex integral functions under qualification conditions

“…x * (t))dµ(t), 16 and so, according to [19,Proposition 3.3] (see also [8,) and Moreau's envelope Theorem,…”