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30101 Pure Mathematics
30102 Applied Mathematics
20103 Numerical and Computational Mathematics
2Convex functions
2Fenchel subdifferential
2Lower semicontinuous Lyapunov pairs and functions
10802 Computation Theory and Mathematics
1Contingent derivatives
1Convex infinite-dimensional
1Differential inclusions
1Evolution differential inclusions
1Fenchel and approximate subdifferentials
1Generalized subdifferentials
1Invariance of sets
1Invariant sets
1Lipschitz perturbations
1Maximal monotone operators
1Maximally monotone operators
1Optimization
1Pointwise supremum function

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- Adly, Samir, Hantoute, Abderrahim, Thera, Michel

**Authors:**Adly, Samir , Hantoute, Abderrahim , Thera, Michel**Date:**2016**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 157, no. 2 (2016), p. 349-374**Full Text:**false**Reviewed:****Description:**The general theory of Lyapunov stability of first-order differential inclusions in Hilbert spaces has been studied by the authors in the previous paper (Adly et al. in Nonlinear Anal 75(3): 985–1008, 2012). This new contribution focuses on the case when the interior of the domain of the maximally monotone operator governing the given differential inclusion is nonempty; this includes in a natural way the finite-dimensional case. The current setting leads to simplified, more explicit criteria and permits some flexibility in the choice of the generalized subdifferentials. Some consequences of the viability of closed sets are given. Our analysis makes use of standard tools from convex and variational analysis. © 2015, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

Towards supremum-sum subdifferential calculus free of qualification conditions

- Correa, Rafael, Hantoute, Abderrahim, Lopez, Marco

**Authors:**Correa, Rafael , Hantoute, Abderrahim , Lopez, Marco**Date:**2016**Type:**Text , Journal article**Relation:**Siam Journal on Optimization Vol. 26, no. 4 (2016), p. 2219-2234**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We give a formula for the subdifferential of the sum of two convex functions where one of them is the supremum of an arbitrary family of convex functions. This is carried out under a weak assumption expressing a natural relationship between the lower semicontinuous envelopes of the data functions in the domain of the sum function. We also provide a new rule for the subdifferential of the sum of two convex functions, which uses a strategy of augmenting the involved functions. The main feature of our analysis is that no continuity-type condition is required. Our approach allows us to unify, recover, and extend different results in the recent literature.

**Authors:**Correa, Rafael , Hantoute, Abderrahim , Lopez, Marco**Date:**2016**Type:**Text , Journal article**Relation:**Siam Journal on Optimization Vol. 26, no. 4 (2016), p. 2219-2234**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We give a formula for the subdifferential of the sum of two convex functions where one of them is the supremum of an arbitrary family of convex functions. This is carried out under a weak assumption expressing a natural relationship between the lower semicontinuous envelopes of the data functions in the domain of the sum function. We also provide a new rule for the subdifferential of the sum of two convex functions, which uses a strategy of augmenting the involved functions. The main feature of our analysis is that no continuity-type condition is required. Our approach allows us to unify, recover, and extend different results in the recent literature.

Valadier-like formulas for the supremum function I

- Correa, Rafael, Hantoute, Abderrahim, López, Marco

**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**Full Text:**false**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.

Weaker conditions for subdifferential calculus of convex functions

- Correa, Rafael, Hantoute, Abderrahim, Lopez, Marco

**Authors:**Correa, Rafael , Hantoute, Abderrahim , Lopez, 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.

Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions

- Adly, Samir, Hantoute, Abderrahim, Théra, Michel

**Authors:**Adly, Samir , Hantoute, Abderrahim , Théra, Michel**Date:**2012**Type:**Text , Journal article**Relation:**Nonlinear Analysis: Theory, Methods & Applications Vol. 75, no. 3 (February, 2012), p. 985-1008**Full Text:**false**Reviewed:****Description:**The main objective of this paper is to provide new explicit criteria to characterize weak lower semicontinuous Lyapunov pairs or functions associated to first-order differential inclusions in Hilbert spaces. These inclusions are governed by a Lipschitzian perturbation of a maximally monotone operator. The dual criteria we give are expressed by means of the proximal and basic subdifferentials of the nominal functions while primal conditions are described in terms of the contingent directional derivative. We also propose a unifying review of many other criteria given in the literature. Our approach is based on advanced tools of variational analysis and generalized differentiation.

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