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2Abstract convexity
2Convex-along-rays functions
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1Best approximation
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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

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

Best approximation by downward sets with applications

- Rubinov, Alex, Mohebi, Hossein

**Authors:**Rubinov, Alex , Mohebi, Hossein**Date:**2006**Type:**Text , Journal article**Relation:**Analysis in Theory and Applications Vol. 22, no. 1 (2006), p. 20-40**Full Text:****Reviewed:****Description:**We develop a theory of downward sets for a class of normed ordered spaces. We study best approximation in a normed ordered space X by elements of downward sets, and give necessary and sufficient conditions for any element of best approximation by a closed downward subset of X. We also characterize strictly downward subsets of X, and prove that a downward subset of X is strictly downward if and only if each its boundary point is Chebyshev. The results obtained are used for examination of some Chebyshev pairs (W,x), where x E X and W is a closed downward subset of X.**Description:**C1**Description:**2003001535

**Authors:**Rubinov, Alex , Mohebi, Hossein**Date:**2006**Type:**Text , Journal article**Relation:**Analysis in Theory and Applications Vol. 22, no. 1 (2006), p. 20-40**Full Text:****Reviewed:****Description:**We develop a theory of downward sets for a class of normed ordered spaces. We study best approximation in a normed ordered space X by elements of downward sets, and give necessary and sufficient conditions for any element of best approximation by a closed downward subset of X. We also characterize strictly downward subsets of X, and prove that a downward subset of X is strictly downward if and only if each its boundary point is Chebyshev. The results obtained are used for examination of some Chebyshev pairs (W,x), where x E X and W is a closed downward subset of X.**Description:**C1**Description:**2003001535

Conical decomposition and vector lattices with respect to several preorders

- Baratov, Rishat, Rubinov, Alex

**Authors:**Baratov, Rishat , Rubinov, Alex**Date:**2006**Type:**Text , Journal article**Relation:**Taiwanese Journal of Mathematics Vol. 10, no. 2 (2006), p. 265-298**Full Text:****Reviewed:****Description:**The decomposition set-valued mapping in a Banach space E with cones K i,i = 1,..., n describes all decompositions of a given element on addends, such that addend i belongs to the i-th cone. We examine the decomposition mapping and its dual. We study conditions that provide the additivity of the decomposition mapping. For this purpose we introduce and study the Riesz interpolation property and lattice properties of spaces with respect to several preorders. The notion of 2-vector lattice is introduced and studied. Theorems that establish the relationship between the Riesz interpolation property and lattice properties of the dual spaces are given.**Description:**C1**Description:**2003001553

**Authors:**Baratov, Rishat , Rubinov, Alex**Date:**2006**Type:**Text , Journal article**Relation:**Taiwanese Journal of Mathematics Vol. 10, no. 2 (2006), p. 265-298**Full Text:****Reviewed:****Description:**The decomposition set-valued mapping in a Banach space E with cones K i,i = 1,..., n describes all decompositions of a given element on addends, such that addend i belongs to the i-th cone. We examine the decomposition mapping and its dual. We study conditions that provide the additivity of the decomposition mapping. For this purpose we introduce and study the Riesz interpolation property and lattice properties of spaces with respect to several preorders. The notion of 2-vector lattice is introduced and studied. Theorems that establish the relationship between the Riesz interpolation property and lattice properties of the dual spaces are given.**Description:**C1**Description:**2003001553

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

Hidden abstract convex functions

- Rubinov, Alex, Wu, Zhiyou, Li, Duan

**Authors:**Rubinov, Alex , Wu, Zhiyou , Li, Duan**Date:**2005**Type:**Text , Journal article**Relation:**Journal of Nonlinear and Convex Analysis Vol. 6, no. 1 (2005), p. 203-216**Full Text:**false**Reviewed:****Description:**C1**Description:**2003001424

Sigma-porosity in monotonic analysis with applications to optimization

**Authors:**Rubinov, Alex**Date:**2005**Type:**Text , Journal article**Relation:**Abstract and Applied Analysis Vol. 2005, no. 3 (2005), p. 287-305**Full Text:**false**Reviewed:****Description:**We introduce and study some metric spaces of increasing positively homogeneous (IPH) functions, decreasing functions, and conormal (upward) sets. We prove that the complements of the subset of strictly increasing IPH functions, of the subset of strictly decreasing functions, and of the subset of strictly conormal sets are $sigma$-porous in corresponding spaces. Some applications to optimization are given.**Description:**C1**Description:**We introduce and study some metric spaces of increasing positively homogeneous (IPH) functions, decreasing functions, and conormal (upward) sets. We prove that the complements of the subset of strictly increasing IPH functions, of the subset of strictly decreasing functions, and of the subset of strictly conormal sets are $\sigma$-porous in corresponding spaces. Some applications to optimization are given.**Description:**2003001421

Hermite-Hadamard-type inequalities for increasing convex-along-rays function

- Rubinov, Alex, Dragomir, S. S, Dutta, J.

**Authors:**Rubinov, Alex , Dragomir, S. S , Dutta, J.**Date:**2004**Type:**Text , Journal article**Relation:**Analysis Vol. 24, no. 2 (2004), p. 171-181**Full Text:**false**Reviewed:****Description:**C1**Description:**2003000933

Lagrange-type functions in constrained optimization

- Rubinov, Alex, Yang, Xiao, Bagirov, Adil, Gasimov, Rafail

**Authors:**Rubinov, Alex , Yang, Xiao , Bagirov, Adil , Gasimov, Rafail**Date:**2003**Type:**Text , Journal article**Relation:**Journal of Mathematical Sciences Vol. 115, no. 4 (2003), p. 2437-2505**Full Text:**false**Reviewed:****Description:**We examine various kinds of nonlinear Lagrange-type functions for constrained optimization problems. In particular, we study the weak duality, the zero duality gap property, and the existence of an exact parameter for these functions. The paper contains a detailed survey of results in these directions and comparison of different methods proposed by different authors. Some new results are also given.**Description:**C1**Description:**2003000358

Attracting sets for increasing co-radiant and topical operators

- Kloeden, Peter, Rubinov, Alex

**Authors:**Kloeden, Peter , Rubinov, Alex**Date:**2002**Type:**Text , Journal article**Relation:**Mathematische Nachrichten Vol. 243, no. (2002), p. 134-145**Full Text:**false**Reviewed:****Description:**A generalization of the Perron-Frobenius theorem to increasing positively homogeneous of degree one operators is extended to increasing co-radiant and topical operators, which are of interest in mathematical economics. In particular, small attracting sets containing the limit points of all sequences generated by iteration of such operators are determined.**Description:**C1**Description:**2003000150

Hadamard type inequality for quasiconvex functions in higher dimensions

**Authors:**Rubinov, Alex , Dutta, J.**Date:**2002**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 270, no. 1 (2002), p. 80-91**Full Text:**false**Reviewed:****Description:**In this article we study a Hadamard type inequality for nonnegative evenly quasiconvex functions. The approach of our study is based on the notion of abstract convexity. We also provide an explicit calculation to evaluate the asymptotically sharp constant associated with the inequality over a unit square in the two-dimensional plane. © 2002 Elsevier Science (USA). All rights reserved.**Description:**2003000149

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.

Typical behaviour in scalar delay differential equations

- Ivanov, Anatoli, Dzalilov, Zari, Rubinov, Alex

**Authors:**Ivanov, Anatoli , Dzalilov, Zari , Rubinov, Alex**Date:**2001**Type:**Text , Journal article**Relation:**Studies of University of Zilina, Mathematical series Vol. 14 , no. 1 (2001), p. 1-10**Full Text:**false**Reviewed:****Description:**C1**Description:**2003002564

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