Monotonic analysis over cones : III
- 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
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- 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
- 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
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- 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
Hidden abstract convex functions
- 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
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- 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
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- 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
- Authors: Rubinov, Alex , Dragomir, S. S , Dutta, J.
- Date: 2004
- Type: Text , Journal article
- Relation: Analysis Vol. 24, no. 2 (2004), p. 171-181
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- Description: C1
- Description: 2003000933
Lagrange-type functions in constrained optimization
- 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
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- 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
- Authors: Kloeden, Peter , Rubinov, Alex
- Date: 2002
- Type: Text , Journal article
- Relation: Mathematische Nachrichten Vol. 243, no. (2002), p. 134-145
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- 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
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- 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
- 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
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- 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
- 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
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- Description: C1
- Description: 2003002564