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

**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.

### 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:

### A unifying approach to robust convex infinite optimization duality

**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.

### 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.

### About intrinsic transversality of pairs of sets

**Authors:**Kruger, Alexander**Date:**2018**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 26, no. 1 (2018), p. 111-142**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The article continues the study of the ‘regular’ arrangement of a collection of sets near a point in their intersection. Such regular intersection or, in other words, transversality properties are crucial for the validity of qualification conditions in optimization as well as subdifferential, normal cone and coderivative calculus, and convergence analysis of computational algorithms. One of the main motivations for the development of the transversality theory of collections of sets comes from the convergence analysis of alternating projections for solving feasibility problems. This article targets infinite dimensional extensions of the intrinsic transversality property introduced recently by Drusvyatskiy, Ioffe and Lewis as a sufficient condition for local linear convergence of alternating projections. Several characterizations of this property are established involving new limiting objects defined for pairs of sets. Special attention is given to the convex case.

### About subtransversality of collections of sets

**Authors:**Kruger, Alexander , Luke, Russell , Thao, Nguyen**Date:**2017**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 25, no. 4 (2017), p. 701-729**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We provide dual sufficient conditions for subtransversality of collections of sets in an Asplund space setting. For the convex case, we formulate a necessary and sufficient dual criterion of subtransversality in general Banach spaces. Our more general results suggest an intermediate notion of subtransversality, what we call weak intrinsic subtransversality, which lies between intrinsic transversality and subtransversality in Asplund spaces.

### Best approximate solutions of inconsistent linear inequality systems

**Authors:**Goberna, Miguel , Hiriart-Urruty, Jean-Baptiste , López, Marco**Date:**2018**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 46, no. 2 (2018), p. 271-284**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper is intended to characterize three types of best approximate solutions for inconsistent linear inequality systems with an arbitrary number of constraints. It also gives conditions guaranteeing the existence of best uniform solutions and discusses potential applications.

### Borwein–Preiss vector variational principle

**Authors:**Kruger, Alexander , Plubtieng, Somyot , Seangwattana, Thidaporn**Date:**2017**Type:**Text , Journal article**Relation:**Positivity Vol. 21, no. 4 (2017), p. 1273-1292**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This article extends to the vector setting the results of our previous work Kruger et al. (J Math Anal Appl 435(2):1183–1193, 2016) which refined and slightly strengthened the metric space version of the Borwein–Preiss variational principle due to Li and Shi (J Math Anal Appl 246(1):308–319, 2000. doi:10.1006/jmaa.2000.6813). We introduce and characterize two seemingly new natural concepts of ε-minimality, one of them dependent on the chosen element in the ordering cone and the fixed “gauge-type” function. © 2017, Springer International Publishing.

### Convexity and closedness in stable robust duality

**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.

### Ekeland's inverse function theorem in graded Fréchet spaces revisited for multifunctions

**Authors:**Huynh, Van Ngai , Théra, Michel**Date:**2018**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 457, no. 2 (2018), p. 1403-1421**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**In this paper, we present some inverse function theorems and implicit function theorems for set-valued mappings between Fréchet spaces. The proof relies on Lebesgue's Dominated Convergence Theorem and on Ekeland's variational principle. An application to the existence of solutions of differential equations in Fréchet spaces with non-smooth data is given.

### Enlargements of the moreau–rockafellar subdifferential

**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

### Franco Giannessi : “Every mathematician can do a true theorem : only a genius can make an important mistake”

**Authors:**Théra, Michel**Date:**2022**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 193, no. 1-3 (2022), p. 5-20**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:**

### Gateaux differentiability revisited

**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.

### Indexation strategies and calmness constants for uncertain linear inequality systems

**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.

### Metric regularity relative to a cone

**Authors:**Van Ngai, Huynh , Tron, Nguyen , Théra, Michel**Date:**2019**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 47, no. 3 (2019), p. 733-756**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The purpose of this paper is to discuss some of the highlights of the theory of metric regularity relative to a cone. For example, we establish a slope and some coderivative characterizations of this concept, as well as some stability results with respect to a Lipschitz perturbation.

### 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.

### Nonlinear transversality of collections of sets : dual space necessary characterizations

**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.

### On (local) analysis of multifunctions via subspaces contained in graphs of generalized derivatives

**Authors:**Gfrerer, Helmut , Outrata, Jiri**Date:**2022**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 508, no. 2 (2022), p.**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper deals with a comprehensive theory of mappings, whose local behavior can be described by means of linear subspaces, contained in the graphs of two (primal and dual) generalized derivatives. This class of mappings includes the graphically Lipschitzian mappings and thus a number of multifunctions, frequently arising in optimization and equilibrium problems. The developed theory makes use of new generalized derivatives, provides us with some calculus rules and reveals a number of interesting connections. In particular, it enables us to construct a modification of the semismooth* Newton method with improved convergence properties and to derive a generalization of Clarke's Inverse Function Theorem to multifunctions together with new efficient characterizations of strong metric (sub)regularity and tilt stability. © 2021 The Author(s)

### On computation of optimal strategies in oligopolistic markets respecting the cost of change

**Authors:**Outrata, Jiri , Valdman, Jan**Date:**2020**Type:**Text , Journal article**Relation:**Mathematical Methods of Operations Research Vol. 92, no. 3 (2020), p. 489-509**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper deals with a class of parameterized equilibrium problems, where the objectives of the players do possess nonsmooth terms. The respective Nash equilibria can be characterized via a parameter-dependent variational inequality of the second kind, whose Lipschitzian stability, under appropriate conditions, is established. This theory is then applied to evolution of an oligopolistic market in which the firms adapt their production strategies to changing input costs, while each change of the production is associated with some “costs of change”. We examine both the Cournot-Nash equilibria as well as the two-level case, when one firm decides to take over the role of the Leader (Stackelberg equilibrium). The impact of costs of change is illustrated by academic examples. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

### On lipschitzian properties of implicit multifunctions

**Authors:**Gfrerer, Helmut , Outrata, Jiri**Date:**2016**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 26, no. 4 (2016), p. 2160-2189**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper is devoted to the development of new sufficient conditions for the calmness and the Aubin property of implicit multifunctions. As the basic tool we employ the directional limiting coderivative which, together with the graphical derivative, enables a fine analysis of the local behavior of the investigated multifunction along relevant directions. For verification of the calmness property, in addition, a new condition has been discovered which parallels the missing implicit function paradigm and permits us to replace the original multifunction by a substantially simpler one. Moreover, as an auxiliary tool, a handy formula for the computation of the directional limiting coderivative of the normal-cone map with a polyhedral set has been derived which perfectly matches the framework of [A. L. Dontchev and R. T. Rockafellar, SIAM J. Optim., 6 (1996), pp. 1087{1105]. All important statements are illustrated by examples. © 2016 Society for Industrial and Applied Mathematics.