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4Rubinov, Alex
2Correa, Rafael
2Hantoute, Abderrahim
2Kruger, Alexander
2López, Marco
1Beer, Gerald
1Burachik, Regina
1Crespi, G. P.
1Cuong, Nguyen
1Cánovas, Maria
1Dutta, J.
1Ginchev, I.
1Hofmann, Karl
1Jeyakumar, Vaithilingam
1Lopez, Marco
1Martinez-Legaz, Juan
1Morris, Sidney
1Parra, Juan
1Rocca, M.
1Sharikov, Evgenii

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90101 Pure Mathematics
3Convex functions
2Abstract convexity
2Fenchel subdifferential
2Normal cone
2Pointwise supremum function
2Star-shaped set
2Subdifferential
2Valadier-like formulas
10103 Numerical and Computational Mathematics
1Alternating projections
1Best approximation
1Bounded linear regularity
1Clarke-Ekeland duality
1Closed subgroup theorem
1Co-normal sets
1Co-radiant sets
1Collections
1Compact index set
1Convergence

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Star-shaped separability with applications

- Rubinov, Alex, Sharikov, Evgenii

**Authors:**Rubinov, Alex , Sharikov, Evgenii**Date:**2006**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 13, no. 3-4 (2006), p. 849-860**Full Text:****Reviewed:****Description:**We discuss the notion of a support collection to a star-shaped set at a certain boundary point and a weak separability of two star-shaped sets. Applications to some problems, including the minimization of a star-shaped distance, are given. © Heldermann Verlag.**Description:**C1**Description:**2003001592

**Authors:**Rubinov, Alex , Sharikov, Evgenii**Date:**2006**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 13, no. 3-4 (2006), p. 849-860**Full Text:****Reviewed:****Description:**We discuss the notion of a support collection to a star-shaped set at a certain boundary point and a weak separability of two star-shaped sets. Applications to some problems, including the minimization of a star-shaped distance, are given. © Heldermann Verlag.**Description:**C1**Description:**2003001592

Convex along lines functions and abstract convexity. Part i

- Crespi, G. P., Ginchev, I., Rocca, M., Rubinov, Alex

**Authors:**Crespi, G. P. , Ginchev, I. , Rocca, M. , Rubinov, Alex**Date:**2007**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 14, no. 1 (2007), p. 185-204**Full Text:**false**Reviewed:****Description:**The present paper investigates the property of a function f : Rn → R+∞ := R U {+∞} with f(0) < +∞ to be Ln-subdifferentiable or Hn-convex. The Ln-subdifferentiability and Hnn-convexity are introduced as in Rubinov [9]. Some refinements of these properties lead to the notions of Ln0-subdifferentiability and Hn0-convexity. Their relation to the convex-along (CAL) functions is underlined in the following theorem proved in the paper (Theorem 5.6): Let the function f : Rn → R+∞ be such that f(0) < +∞ and f is Hn-convex at the points at which it is infinite. Then if f is Ln0-subdifferentiable, it is CAL and globally calm at each x0 ∈ dom f. Here the notions of local and global calmness are introduced after Rockafellar, Wets [8] and play an important role in the considerations. The question is posed for the possible reversal of this result. In the case of a positively homogeneous (PH) and CAL function such a reversal is proved (Theorem 6.2). As an application conditions are obtained under which a CAL PH function is Hn0-convex (Theorems 6.3 and 6.4). © Heldermann Verlag.**Description:**C1

Dynamics of positive multiconvex relations

- Vladimirov, Alexander, Rubinov, Alex

**Authors:**Vladimirov, Alexander , Rubinov, Alex**Date:**2001**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 8, no. 2 (2001), p. 387-399**Full Text:**false**Reviewed:****Description:**A notion of multiconvex relation as a union of a finite number of convex relations is introduced. For a particular case of multiconvex process, that is, a union of a finite set of convex processes, we define the notions of the joint and the generalized spectral radius in the same manner as for matrices. We prove the equivalence of these two values if all component processes are positive, bounded, and closed. © Heldermann Verlag.

Monotonic analysis over cones : III

- Dutta, J., Martinez-Legaz, Juan, Rubinov, Alex

**Authors:**Dutta, J. , Martinez-Legaz, Juan , Rubinov, Alex**Date:**2008**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 15, no. 3 (2008), p. 561-579**Full Text:**false**Reviewed:****Description:**This paper studies the class of increasing and co-radiant (ICR) functions over a cone equipped with an order relation which agrees with the conic structure. In particular, a representation of ICR functions as abstract convex functions is provided. This representation suggests the introduction of some polarity notions between sets. The relationship between ICR functions and increasing positively homogeneous functions is also shown.**Description:**C1

On the pro-lie group theorem and the closed subgroup theorem

- Hofmann, Karl, Morris, Sidney

**Authors:**Hofmann, Karl , Morris, Sidney**Date:**2008**Type:**Text , Journal article**Relation:**Journal of Lie Theory Vol. 18, no. 2 (2008), p. 383-390**Full Text:**false**Reviewed:****Description:**Let H and M be closed normal subgroups of a pro-Lie group G and assume that H is connected and that G/M is a Lie group. Then there is a closed normal subgroup N of G such that N ? M, that G/N is a Lie group, and that HN is closed in G. As a consequence, H/(H ? N) ? HN/N is an isomorphism of Lie groups. © 2008 Heldermann Verlag.**Description:**C1

A dual condition for the convex subdifferential sum formula with applications

- Burachik, Regina, Jeyakumar, Vaithilingam

**Authors:**Burachik, Regina , Jeyakumar, Vaithilingam**Date:**2005**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 12, no. 2 (2005), p. 279-290**Full Text:**false**Reviewed:****Description:**C1**Description:**2003002555

Regularity of collections of sets and convergence of inexact alternating projections

- Kruger, Alexander, Thao, Nguyen

**Authors:**Kruger, Alexander , Thao, Nguyen**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 23, no. 3 (2016), p. 823-847**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:**false**Reviewed:****Description:**We study the usage of regularity properties of collections of sets in convergence analysis of alternating projection methods for solving feasibility problems. Several equivalent characterizations of these properties are provided. Two settings of inexact alternating projections are considered and the corresponding convergence estimates are established and discussed.

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.

Valadier-like Formulas for the Supremum Function II: The Compactly Indexed Case

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

**Authors:**Correa, Rafael , Hantoute, Abderrahim , Lopez, Marco**Date:**2019**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 26, no. 1 (2019), p. 299-324**Full Text:**false**Reviewed:****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.

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.

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:

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