Weaker conditions for subdifferential calculus of convex functions
- Authors: Correa, Rafael , Hantoute, Abderrahim , López, Marco
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Functional Analysis Vol. 271, no. 5 (2016), p. 1177-1212
- Relation: http://purl.org/au-research/grants/arc/DP160100854
- Full Text: false
- Reviewed:
- Description: 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.
- Description: 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. (C) 2016 Elsevier Inc. All rights reserved.
Valadier-like formulas for the supremum function I
- Authors: Correa, Rafael , Hantoute, Abderrahim , López, Marco
- Date: 2018
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 25, no. 4 (2018), p. 1253-1278
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Reviewed:
- Description: We generalize and improve the original characterization given by Valadier [19, Theorem 1] of the subdifferential of the pointwise supremum of convex functions, involving the subdifferentials of the data functions at nearby points. We remove the continuity assumption made in that work and obtain a general formula for such a subdifferential. In particular, when the supremum is continuous at some point of its domain, but not necessarily at the reference point, we get a simpler version which gives rise to the Valadier formula. Our starting result is the characterization given in [11, Theorem 4], which uses the e-subdifferential at the reference point.
A uniform approach to hölder calmness of subdifferentials
- Authors: Beer, Gerald , Cánovas, Maria , López, Marco , Parra, Juan
- Date: 2020
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 27, no. 1 (2020), p.
- Relation: http://purl.org/au-research/grants/arc/DP160100854
- Full Text: false
- Reviewed:
- Description: For finite-valued convex functions f defined on the n-dimensional Euclidean space, we are interested in the set-valued mapping assigning to each pair (f, x) the subdifferential of f at x. Our approach is uniform with respect to f in the sense that it involves pairs of functions close enough to each other, but not necessarily around a nominal function. More precisely, we provide lower and upper estimates, in terms of Hausdorff excesses, of the subdifferential of one of such functions at a nominal point in terms of the subdifferential of nearby functions in a ball centered in such a point. In particular, we obtain the (1/2) - Hölder calmness of our mapping at a nominal pair (f, x) under the assumption that the subdifferential mapping viewed as a set-valued mapping from Rn to Rn with f fixed is calm at each point of {x} × ∂f(x). © Heldermann Verlag
- Description: Funding details: Australian Research Council, ARC, DP160100854 Funding details: European Commission, EU Funding details: Ministerio de Economía y Competitividad, MINECO Funding details: Federación Española de Enfermedades Raras, FEDER Funding text 1: