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Abstract convexity for nonconvex optimization duality

- Nedic, A., Ozdaglar, A., Rubinov, Alex

**Authors:**Nedic, A. , Ozdaglar, A. , Rubinov, Alex**Date:**2007**Type:**Text , Journal article**Relation:**Optimization Vol. 56, no. 5-6 (2007), p. 655-674**Full Text:**false**Reviewed:****Description:**In this article, we use abstract convexity results to study augmented dual problems for (nonconvex) constrained optimization problems. We consider a nonincreasing function f that is lower semicontinuous at 0 and establish its abstract convexity at 0 with respect to a set of elementary functions defined by nonconvex augmenting functions. We consider three different classes of augmenting functions: nonnegative augmenting functions, bounded-below augmenting functions, and unbounded augmenting functions. We use the abstract convexity results to study augmented optimization duality without imposing boundedness assumptions.**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

Increasing quasiconcave co-radiant functions with applications in mathematical economics

- Martinez-Legaz, Juan, Rubinov, Alex, Schaible, Siegfried

**Authors:**Martinez-Legaz, Juan , Rubinov, Alex , Schaible, Siegfried**Date:**2005**Type:**Text , Journal article**Relation:**Mathematical Methods of Operations Research Vol. 61, no. 2 (2005), p. 261-280**Full Text:**false**Reviewed:****Description:**We study increasing quasiconcave functions which are co-radiant. Such functions have frequently been employed in microeconomic analysis. The study is carried out in the contemporary framework of abstract convexity and abstract concavity. Various properties of these functions are derived. In particular we identify a small "natural" infimal generator of the set of all coradiant quasiconcave increasing functions. We use this generator to examine two duality schemes for these functions: classical duality often used in microeconomic analysis and a more recent duality concept. Some possible applications to the theory of production functions and utility functions are discussed. © Springer-Verlag 2005.**Description:**C1**Description:**2003001423

On the absence of duality gap for Lagrange-type functions

- Rubinov, Alex, Burachik, Regina

**Authors:**Rubinov, Alex , Burachik, Regina**Date:**2005**Type:**Text , Journal article**Relation:**Journal of Industrial and Management Optimization Vol. 1, no. 1 (2005), p. 33-38**Full Text:**false**Reviewed:****Description:**Given a generic dual program we discuss the absence of duality gap for a family of Lagrange-type functions. We obtain necessary conditions that become sufficient ones under some additional assumptions. We also give examples of Lagrangetype functions for which this sufficient conditions hold.**Description:**C1**Description:**2003001425

- Rubinov, Alex, Gasimov, Rafail

**Authors:**Rubinov, Alex , Gasimov, Rafail**Date:**2004**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 29, no. 4 (2004), p. 455-477**Full Text:**false**Reviewed:****Description:**We consider problems of vector optimization with preferences that are not necessarily a pre-order relation. We introduce the class of functions which can serve for a scalarization of these problems and consider a scalar duality based on recently developed methods for non-linear penalization scalar problems with a single constraint.**Description:**C1**Description:**2003000932

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

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