Description:
The theory of abstract convexity generalizes ideas of convex analysis by using the notion of global supports and the global definition of subdifferential. In order to apply this theory to optimization, we need to extend subdifferential calculus and separation properties into the area of abstract convexity.
Description:
The theory of abstract convexity generalizes ideas of convex analysis by using the notion of global supports and the global definition of subdifferential. In order to apply this theory to optimization, we need to extend subdifferential calculus and separation properties into the area of abstract convexity.
Description:
The main part of this thesis presents a new approach to the topic of conjugation, with applications to various optimization problems. It does so by introducing (what we call) G-coupling functions.
Description:
The main part of this thesis presents a new approach to the topic of conjugation, with applications to various optimization problems. It does so by introducing (what we call) G-coupling functions.
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.
Description:
Continuing with the work on the subdifferential of the pointwise supremum of convex functions, started in part I of this paper [R. Correa, A. Hantoute, M. A. Lopez, Valadier-like formulas for the supremum function I, J. Convex Analysis 25 (2018) 1253-1278], we focus now on the compactly indexed case. We assume that the index set is compact and that the data functions are upper semicontinuous with respect to the index variable (actually, this assumption will only affect the set of epsilon-active indices at the reference point). As in the previous work, we do not require any continuity assumption with respect to the decision variable. The current compact setting gives rise to more explicit formulas, which only involve subdifferentials at the reference point of active data functions. Other formulas are derived under weak continuity assumptions. These formulas reduce to the characterization given by M. Valadier [Sous-differentiels d'une borne superieure et d'une somme continue de fonctions convexes, C. R. Acad. Sci. Paris Ser. A-B Math. 268 (1969) 39-42, Theorem 2], when the supremum function is continuous.
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.
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.
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: