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15Kruger, Alexander
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4Gfrerer, Helmut
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3Hantoute, Abderrahim
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40802 Computation Theory and Mathematics
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Enlargements of the moreau–rockafellar subdifferential

- Abbasi, Malek, Kruger, Alexander, Théra, Michel

**Authors:**Abbasi, Malek , Kruger, Alexander , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 29, no. 3 (2021), p. 701-719**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper proposes three enlargements of the conventional Moreau–Rockafellar subdifferential: the sup-, sup

**Authors:**Abbasi, Malek , Kruger, Alexander , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 29, no. 3 (2021), p. 701-719**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper proposes three enlargements of the conventional Moreau–Rockafellar subdifferential: the sup-, sup

Gateaux differentiability revisited

- Abbasi, Malek, Kruger, Alexander, Théra, Michel

**Authors:**Abbasi, Malek , Kruger, Alexander , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Applied Mathematics and Optimization Vol. 84, no. 3 (2021), p. 3499-3516**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We revisit some basic concepts and ideas of the classical differential calculus and convex analysis extending them to a broader frame. We reformulate and generalize the notion of Gateaux differentiability and propose new notions of generalized derivative and generalized subdifferential in an arbitrary topological vector space. Meaningful examples preserving the key properties of the original notion of derivative are provided. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.

**Authors:**Abbasi, Malek , Kruger, Alexander , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Applied Mathematics and Optimization Vol. 84, no. 3 (2021), p. 3499-3516**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We revisit some basic concepts and ideas of the classical differential calculus and convex analysis extending them to a broader frame. We reformulate and generalize the notion of Gateaux differentiability and propose new notions of generalized derivative and generalized subdifferential in an arbitrary topological vector space. Meaningful examples preserving the key properties of the original notion of derivative are provided. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.

Set-valued orthogonality and nearness

- Barbagallo, Annamaria, Ernst, Octavian, Théra, Michel

**Authors:**Barbagallo, Annamaria , Ernst, Octavian , Théra, Michel**Date:**2020**Type:**Text , Journal article**Relation:**AAPP Atti della Accademia Peloritana dei Pericolanti, Classe di Scienze Fisiche, Matematiche e Naturali Vol. 98, no. (2020), p.**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The theory of set-valued mappings has grown with the development of modern variational analysis. It is a key in convex and non-smooth analysis, in game theory, in mathematical economics and in control theory. The concepts of nearness and orthogonality have been known for functions since the pioneering works of Campanato, Birkhoff and James. In a recent paper Barbagallo et al. [J. Math. Anal. Appl., 484 (1), (2020)] a connection between these two concepts has been made. This note is mainly devoted to introduce nearness and orthogonality between set-valued mappings with the goal to study the solvability of generalized equations involving set-valued mappings. © 2020 Accademia Peloritana dei Pericolanti. All rights reserved.

**Authors:**Barbagallo, Annamaria , Ernst, Octavian , Théra, Michel**Date:**2020**Type:**Text , Journal article**Relation:**AAPP Atti della Accademia Peloritana dei Pericolanti, Classe di Scienze Fisiche, Matematiche e Naturali Vol. 98, no. (2020), p.**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The theory of set-valued mappings has grown with the development of modern variational analysis. It is a key in convex and non-smooth analysis, in game theory, in mathematical economics and in control theory. The concepts of nearness and orthogonality have been known for functions since the pioneering works of Campanato, Birkhoff and James. In a recent paper Barbagallo et al. [J. Math. Anal. Appl., 484 (1), (2020)] a connection between these two concepts has been made. This note is mainly devoted to introduce nearness and orthogonality between set-valued mappings with the goal to study the solvability of generalized equations involving set-valued mappings. © 2020 Accademia Peloritana dei Pericolanti. All rights reserved.

On the regularity of weak solutions of the boussinesq equations in besov spaces

- Barbagallo, Annamaria, Gala, Sadek, Ragusa, Maria, Théra, Michel

**Authors:**Barbagallo, Annamaria , Gala, Sadek , Ragusa, Maria , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 49, no. 3 (2021), p. 637-649**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The main issue addressed in this paper concerns an extension of a result by Z. Zhang who proved, in the context of the homogeneous Besov space Ḃ

**Authors:**Barbagallo, Annamaria , Gala, Sadek , Ragusa, Maria , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 49, no. 3 (2021), p. 637-649**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The main issue addressed in this paper concerns an extension of a result by Z. Zhang who proved, in the context of the homogeneous Besov space Ḃ

A uniform approach to hölder calmness of subdifferentials

- Beer, Gerald, Cánovas, Maria, López, Marco, Parra, Juan

**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:

About extensions of the extremal principle

**Authors:**Bui, Hoa , Kruger, Alexander**Date:**2018**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 46, no. 2 (2018), p. 215-242**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**In this paper, after recalling and discussing the conventional extremality, local extremality, stationarity and approximate stationarity properties of collections of sets, and the corresponding (extended) extremal principle, we focus on extensions of these properties and the corresponding dual conditions with the goal to refine the main arguments used in this type of results, clarify the relationships between different extensions, and expand the applicability of the generalized separation results. We introduce and study new more universal concepts of relative extremality and stationarity and formulate the relative extended extremal principle. Among other things, certain stability of the relative approximate stationarity is proved. Some links are established between the relative extremality and stationarity properties of collections of sets and (the absence of) certain regularity, lower semicontinuity, and Lipschitz-like properties of set-valued mappings. © 2018, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.

**Authors:**Bui, Hoa , Kruger, Alexander**Date:**2018**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 46, no. 2 (2018), p. 215-242**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**In this paper, after recalling and discussing the conventional extremality, local extremality, stationarity and approximate stationarity properties of collections of sets, and the corresponding (extended) extremal principle, we focus on extensions of these properties and the corresponding dual conditions with the goal to refine the main arguments used in this type of results, clarify the relationships between different extensions, and expand the applicability of the generalized separation results. We introduce and study new more universal concepts of relative extremality and stationarity and formulate the relative extended extremal principle. Among other things, certain stability of the relative approximate stationarity is proved. Some links are established between the relative extremality and stationarity properties of collections of sets and (the absence of) certain regularity, lower semicontinuity, and Lipschitz-like properties of set-valued mappings. © 2018, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.

Necessary conditions for non-intersection of collections of sets

**Authors:**Bui, Hoa , Kruger, Alexander**Date:**2022**Type:**Text , Journal article**Relation:**Optimization Vol. 71, no. 1 (2022), p. 165-196**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper continues studies of non-intersection properties of finite collections of sets initiated 40 years ago by the extremal principle. We study elementary non-intersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish new primal (slope) and dual (generalized separation) necessary conditions for these non-intersection properties. The results are applied to convergence analysis of alternating projections. © 2021 Informa UK Limited, trading as Taylor & Francis Group.

**Authors:**Bui, Hoa , Kruger, Alexander**Date:**2022**Type:**Text , Journal article**Relation:**Optimization Vol. 71, no. 1 (2022), p. 165-196**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper continues studies of non-intersection properties of finite collections of sets initiated 40 years ago by the extremal principle. We study elementary non-intersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish new primal (slope) and dual (generalized separation) necessary conditions for these non-intersection properties. The results are applied to convergence analysis of alternating projections. © 2021 Informa UK Limited, trading as Taylor & Francis Group.

Strong metric subregularity of mappings in variational analysis and optimization

- Cibulka, Radek, Dontchev, Asen, Kruger, Alexander

**Authors:**Cibulka, Radek , Dontchev, Asen , Kruger, Alexander**Date:**2018**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 457, no. 2 (2018), p. 1247-1287**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**Although the property of strong metric subregularity of set-valued mappings has been present in the literature under various names and with various (equivalent) definitions for more than two decades, it has attracted much less attention than its older “siblings”, the metric regularity and the strong (metric) regularity. The purpose of this paper is to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications. We show that the strong metric subregularity of a mapping F acting between metric spaces is stable under perturbations of the form f+F, where f is a function with a small calmness constant. This result is parallel to the Lyusternik–Graves theorem for metric regularity and to the Robinson theorem for strong regularity, where the perturbations are represented by a function f with a small Lipschitz constant. Then we study perturbation stability of the same kind for mappings acting between Banach spaces, where f is not necessarily differentiable but admits a set-valued derivative-like approximation. Strong metric q-subregularity is also considered, where q is a positive real constant appearing as exponent in the definition. Rockafellar's criterion for strong metric subregularity involving injectivity of the graphical derivative is extended to mappings acting in infinite-dimensional spaces. A sufficient condition for strong metric subregularity is established in terms of surjectivity of the Fréchet coderivative, and it is shown by a counterexample that surjectivity of the limiting coderivative is not a sufficient condition for this property, in general. Then various versions of Newton's method for solving generalized equations are considered including inexact and semismooth methods, for which superlinear convergence is shown under strong metric subregularity. As applications to optimization, a characterization of the strong metric subregularity of the KKT mapping is obtained, as well as a radius theorem for the optimality mapping of a nonlinear programming problem. Finally, an error estimate is derived for a discrete approximation in optimal control under strong metric subregularity of the mapping involved in the Pontryagin principle.

On semiregularity of mappings

- Cibulka, Radek, Fabian, Marian, Kruger, Alexander

**Authors:**Cibulka, Radek , Fabian, Marian , Kruger, Alexander**Date:**2019**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 473, no. 2 (2019), p. 811-836**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**There are two basic ways of weakening the definition of the well-known metric regularity property by fixing one of the points involved in the definition. The first resulting property is called metric subregularity and has attracted a lot of attention during the last decades. On the other hand, the latter property which we call semiregularity can be found under several names and the corresponding results are scattered in the literature. We provide a self-contained material gathering and extending the existing theory on the topic. We demonstrate a clear relationship with other regularity properties, for example, the equivalence with the so-called openness with a linear rate at the reference point is shown. In particular cases, we derive necessary and/or sufficient conditions of both primal and dual type. We illustrate the importance of semiregularity in the convergence analysis of an inexact Newton-type scheme for generalized equations with not necessarily differentiable single-valued part. © 2019 Elsevier Inc.

**Authors:**Cibulka, Radek , Fabian, Marian , Kruger, Alexander**Date:**2019**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 473, no. 2 (2019), p. 811-836**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**There are two basic ways of weakening the definition of the well-known metric regularity property by fixing one of the points involved in the definition. The first resulting property is called metric subregularity and has attracted a lot of attention during the last decades. On the other hand, the latter property which we call semiregularity can be found under several names and the corresponding results are scattered in the literature. We provide a self-contained material gathering and extending the existing theory on the topic. We demonstrate a clear relationship with other regularity properties, for example, the equivalence with the so-called openness with a linear rate at the reference point is shown. In particular cases, we derive necessary and/or sufficient conditions of both primal and dual type. We illustrate the importance of semiregularity in the convergence analysis of an inexact Newton-type scheme for generalized equations with not necessarily differentiable single-valued part. © 2019 Elsevier Inc.

Weaker conditions for subdifferential calculus of convex functions

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

**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.

Towards supremum-sum subdifferential calculus free of qualification conditions

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

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

**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:****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.

Nonlinear transversality of collections of sets : dual space necessary characterizations

- Cuong, Nguyen, Kruger, Alexander

**Authors:**Cuong, Nguyen , Kruger, Alexander**Date:**2020**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 27, no. 1 (2020), p. 285-306**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**This paper continues the study of `good arrangements' of collections of sets in normed spaces near a point in their intersection. Our aim is to study general nonlinear transversality properties. We focus on dual space (subdifferential and normal cone) necessary characterizations of these properties. As an application, we provide dual necessary conditions for the nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe.**Description:**The research was supported by the Australian Research Council, project DP160100854. The second author benefited from the support of the FMJH Program PGMO and from the support of EDF.

Transversality properties : primal sufficient conditions

- Cuong, Nguyen, Kruger, Alexander

**Authors:**Cuong, Nguyen , Kruger, Alexander**Date:**2021**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 29, no. 2 (2021), p. 221-256**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper studies ‘good arrangements’ (transversality properties) of collections of sets in a normed vector space near a given point in their intersection. We target primal (metric and slope) characterizations of transversality properties in the nonlinear setting. The Hölder case is given a special attention. Our main objective is not formally extending our earlier results from the Hölder to a more general nonlinear setting, but rather to develop a general framework for quantitative analysis of transversality properties. The nonlinearity is just a simple setting, which allows us to unify the existing results on the topic. Unlike the well-studied subtransversality property, not many characterizations of the other two important properties: semitransversality and transversality have been known even in the linear case. Quantitative relations between nonlinear transversality properties and the corresponding regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe are also discussed. © 2020, Springer Nature B.V.

**Authors:**Cuong, Nguyen , Kruger, Alexander**Date:**2021**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 29, no. 2 (2021), p. 221-256**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper studies ‘good arrangements’ (transversality properties) of collections of sets in a normed vector space near a given point in their intersection. We target primal (metric and slope) characterizations of transversality properties in the nonlinear setting. The Hölder case is given a special attention. Our main objective is not formally extending our earlier results from the Hölder to a more general nonlinear setting, but rather to develop a general framework for quantitative analysis of transversality properties. The nonlinearity is just a simple setting, which allows us to unify the existing results on the topic. Unlike the well-studied subtransversality property, not many characterizations of the other two important properties: semitransversality and transversality have been known even in the linear case. Quantitative relations between nonlinear transversality properties and the corresponding regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe are also discussed. © 2020, Springer Nature B.V.

Primal necessary characterizations of transversality properties

- Cuong, Nguyen, Kruger, Alexander

**Authors:**Cuong, Nguyen , Kruger, Alexander**Date:**2021**Type:**Text , Journal article**Relation:**Positivity Vol. 25, no. 2 (2021), p. 531-558**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper continues the study of general nonlinear transversality properties of collections of sets and focuses on primal necessary (in some cases also sufficient) characterizations of the properties. We formulate geometric, metric and slope characterizations, particularly in the convex setting. The Hölder case is given a special attention. Quantitative relations between the nonlinear transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings as well as two nonlinear transversality properties of a convex set-valued mapping to a convex set in the range space are discussed. © 2020, Springer Nature Switzerland AG.

**Authors:**Cuong, Nguyen , Kruger, Alexander**Date:**2021**Type:**Text , Journal article**Relation:**Positivity Vol. 25, no. 2 (2021), p. 531-558**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper continues the study of general nonlinear transversality properties of collections of sets and focuses on primal necessary (in some cases also sufficient) characterizations of the properties. We formulate geometric, metric and slope characterizations, particularly in the convex setting. The Hölder case is given a special attention. Quantitative relations between the nonlinear transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings as well as two nonlinear transversality properties of a convex set-valued mapping to a convex set in the range space are discussed. © 2020, Springer Nature Switzerland AG.

Indexation strategies and calmness constants for uncertain linear inequality systems

- Cánovas, Maria, Henrion, René, López, Marco, Parra, Juan

**Authors:**Cánovas, Maria , Henrion, René , López, Marco , Parra, Juan**Date:**2018**Type:**Text , Book chapter**Relation:**The Mathematics of the Uncertain (part of the Studies in Systems, Decision and Control series) p. 831-843**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**The present paper deals with uncertain linear inequality systems viewed as nonempty closed coefficient sets in the (n+ 1) -dimensional Euclidean space. The perturbation size of these uncertainty sets is measured by the (extended) Hausdorff distance. We focus on calmness constants—and their associated neighborhoods—for the feasible set mapping at a given point of its graph. To this aim, the paper introduces an appropriate indexation function which allows us to provide our aimed calmness constants through their counterparts in the setting of linear inequality systems with a fixed index set, where a wide background exists in the literature.

Convexity and closedness in stable robust duality

- Dinh, Nguyen, Goberna, Miguel, López, Marco, Volle, Michel

**Authors:**Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel**Date:**2019**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 13, no. 2 (2019), p. 325-339**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper deals with optimization problems with uncertain constraints and linear perturbations of the objective function, which are associated with given families of perturbation functions whose dual variable depends on the uncertainty parameters. More in detail, the paper provides characterizations of stable strong robust duality and stable robust duality under convexity and closedness assumptions. The paper also reviews the classical Fenchel duality of the sum of two functions by considering a suitable family of perturbation functions.

**Authors:**Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel**Date:**2019**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 13, no. 2 (2019), p. 325-339**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper deals with optimization problems with uncertain constraints and linear perturbations of the objective function, which are associated with given families of perturbation functions whose dual variable depends on the uncertainty parameters. More in detail, the paper provides characterizations of stable strong robust duality and stable robust duality under convexity and closedness assumptions. The paper also reviews the classical Fenchel duality of the sum of two functions by considering a suitable family of perturbation functions.

A unifying approach to robust convex infinite optimization duality

- Dinh, Nguyen, Goberna, Miguel, López, Marco, Volle, Michel

**Authors:**Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 174, no. 3 (2017), p. 650-685**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper considers an uncertain convex optimization problem, posed in a locally convex decision space with an arbitrary number of uncertain constraints. To this problem, where the uncertainty only affects the constraints, we associate a robust (pessimistic) counterpart and several dual problems. The paper provides corresponding dual variational principles for the robust counterpart in terms of the closed convexity of different associated cones.

**Authors:**Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 174, no. 3 (2017), p. 650-685**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper considers an uncertain convex optimization problem, posed in a locally convex decision space with an arbitrary number of uncertain constraints. To this problem, where the uncertainty only affects the constraints, we associate a robust (pessimistic) counterpart and several dual problems. The paper provides corresponding dual variational principles for the robust counterpart in terms of the closed convexity of different associated cones.

On some open problems in optimal control

**Authors:**Dontchev, Asen**Date:**2018**Type:**Text , Book chapter**Relation:**Control Systems and Mathematical Methods in Economics : Essays in Honor of Vladimir M. Veliov (part of the Lecture Notes in Economics and Mathematical Systems book series) p. 3-13**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**Several open problems are presented concerning regularity properties of solutions of optimal control problems with constraints.

A new regularity criterion of weak solutions to the 3D micropolar fluid flows in terms of the pressure

- Gala, Sadek, Ragusa, Maria, Théra, Michel

**Authors:**Gala, Sadek , Ragusa, Maria , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Bolletino dell Unione Matematica Italiana Vol. 14, no. 2 (2021), p. 331-337**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**In this study, we establish a new regularity criterion of weak solutions to the three-dimensional micropolar fluid flows by imposing a critical growth condition on the pressure field. © 2020, Unione Matematica Italiana.

**Authors:**Gala, Sadek , Ragusa, Maria , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Bolletino dell Unione Matematica Italiana Vol. 14, no. 2 (2021), p. 331-337**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**In this study, we establish a new regularity criterion of weak solutions to the three-dimensional micropolar fluid flows by imposing a critical growth condition on the pressure field. © 2020, Unione Matematica Italiana.

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