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14Bagirov, Adil
11Ugon, Julien
9Yearwood, John
7Wu, Zhiyou
7Yang, Xiao
6Dzalilov, Zari
6Kouhbor, Shahnaz
6Kruger, Alexander
6Mammadov, Musa
5Dutta, J.
5Gasimov, Rafail
5Martinez-Legaz, Juan
5Ouveysi, Iradj
5Sukhorukova, Nadezda
4Jeyakumar, Vaithilingam
3Burachik, Regina
3Mohebi, Hossein
3Zhang, Jiapu
2Branch, Philip

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230103 Numerical and Computational Mathematics
23Optimisation
180802 Computation Theory and Mathematics
140102 Applied Mathematics
130101 Pure Mathematics
12Abstract convexity
9Global optimization
8Functions
8Problem solving
6Duality
5Numerical methods
4Augmented Lagrangian
4Cutting angle method
4Global optimisation
4Lagrange multipliers
4Lagrange-type functions
4Normal sets
4Vectors
4WLAN
4Wireless

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On augmented lagrangians for optimization problems with a single constraint

- Gasimov, Rafail, Rubinov, Alex

**Authors:**Gasimov, Rafail , Rubinov, Alex**Date:**2004**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 28, no. 2 (2004), p. 153-173**Full Text:**false**Reviewed:****Description:**We examine augmented Lagrangians for optimization problems with a single (either inequality or equality) constraint. We establish some links between augmented Lagrangians and Lagrange-type functions and propose a new kind of Lagrange-type functions for a problem with a single inequality constraint. Finally, we discuss a supergradient algorithm for calculating optimal values of dual problems corresponding to some class of augmented Lagrangians.**Description:**C1**Description:**2003000929

- Wu, Zhiyou, Jeyakumar, Vaithilingam, Rubinov, Alex

**Authors:**Wu, Zhiyou , Jeyakumar, Vaithilingam , Rubinov, Alex**Date:**2007**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 133, no. 1 (2007), p. 123-130**Full Text:**false**Reviewed:****Description:**We present sufficient conditions for the global optimality of bivalent nonconvex quadratic programs involving quadratic inequality constraints as well as equality constraints. By employing the Lagrangian function, we extend the global subdifferential approach, developed recently in Jeyakumar et al. (J. Glob. Optim., 2007, to appear; Math. Program. Ser. A, 2007, to appear) for studying bivalent quadratic programs without quadratic constraints, and derive global optimality conditions. © 2007 Springer Science+Business Media, LLC.**Description:**C1

Global optimization of marginal functions with applications to economic equilibrium

- Bagirov, Adil, Rubinov, Alex

**Authors:**Bagirov, Adil , Rubinov, Alex**Date:**2001**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 20, no. 3-4 (Aug 2001), p. 215-237**Full Text:**false**Reviewed:****Description:**We discuss the applicability of the cutting angle method to global minimization of marginal functions. The search of equilibrium prices in the exchange model can be reduced to the global minimization of certain functions, which include marginal functions. This problem has been approximately solved by the cutting angle method. Results of numerical experiments are presented and discussed.

Stability of the lower level sets of ICAR functions

- López, Marco, Rubinov, Alex, Vera De Serio, Virginia

**Authors:**López, Marco , Rubinov, Alex , Vera De Serio, Virginia**Date:**2005**Type:**Text , Journal article**Relation:**Numerical Functional Analysis and Optimization Vol. 26, no. 1 (2005), p. 113-127**Full Text:**false**Reviewed:****Description:**In this paper, we study the stability of the lower level set {x E R++n | f (x) ≤ 0} of a finite valued increasing convex-along-rays (ICAR) function f defined on R++n. In monotonic analysis, ICAR functions play the role of usual convex functions in classical convex analysis. We show that each ICAR function f is locally Lipschitz on int dom f and that the pointwise convergence of a sequence of ICAR functions implies its uniform convergence on each compact subset of R ++n. The latter allows us to establish stability results for ICAR functions in some sense similar to those for convex functions. Copyright © Taylor & Francis, Inc.**Description:**C1**Description:**2003001419

A method of truncated codifferential with application to some problems of cluster analysis

- Demyanov, Vladimir, Bagirov, Adil, Rubinov, Alex

**Authors:**Demyanov, Vladimir , Bagirov, Adil , Rubinov, Alex**Date:**2002**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 23, no. 1 (May 2002), p. 63-80**Full Text:**false**Reviewed:****Description:**A method of truncated codifferential descent for minimizing continuously codifferentiable functions is suggested. The convergence of the method is studied. Results of numerical experiments are presented. Application of the suggested method for the solution of some problems of cluster analysis are discussed. In numerical experiments Wisconsin Diagnostic Breast Cancer database was used.**Description:**2003000062

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

Strictly increasing positively homogeneous functions with application to exact penalization

- Rubinov, Alex, Gasimov, Rafail

**Authors:**Rubinov, Alex , Gasimov, Rafail**Date:**2003**Type:**Text , Journal article**Relation:**Optimization Vol. 52, no. 1 (2003), p. 1-28**Full Text:**false**Reviewed:****Description:**We study a nonlinear exact penalization for optimization problems with a single constraint. The penalty function is constructed as a convolution of the objective function and the constraint by means of increasing positively homogeneous (IPH) functions. The main results are obtained for penalization by strictly IPH functions. We show that some restrictive assumptions, which have been made in earlier researches on this topic, can be removed. We also compare the least exact penalty parameters for penalization by different convolution functions. These results are based on some properties of strictly IPH functions that are established in the article.**Description:**C1**Description:**2003000357

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

Dynamical systems based on a fuzzy derivative and its applications to data classification

- Mammadov, Musa, Rubinov, Alex, Yearwood, John

**Authors:**Mammadov, Musa , Rubinov, Alex , Yearwood, John**Date:**2003**Type:**Text , Conference paper**Relation:**Paper presented at the Industrial Optimisation 2003 Conference, Perth : 30th September, 2002**Full Text:**false**Reviewed:****Description:**E1**Description:**2003000339

Minimization of the sum of minima of convex functions and its application to clustering

- Rubinov, Alex, Soukhoroukova, Nadejda, Ugon, Julien

**Authors:**Rubinov, Alex , Soukhoroukova, Nadejda , Ugon, Julien**Date:**2005**Type:**Text , Book chapter**Relation:**Continuous Optimization Chapter p. 409-434**Full Text:****Description:**We study functions that can be represented as the sum of minima of convex functions. Minimization of such functions can be used for approximation of finite sets and their clustering. We suggest to use the local discrete gradient (DG) method [Bag99] and the hybrid method between the cutting angle method and the discrete gradient method (DG+CAM) [BRZ05b] for the minimization of these functions. We report and analyze the results of numerical experiments.**Description:**2003004082

**Authors:**Rubinov, Alex , Soukhoroukova, Nadejda , Ugon, Julien**Date:**2005**Type:**Text , Book chapter**Relation:**Continuous Optimization Chapter p. 409-434**Full Text:****Description:**We study functions that can be represented as the sum of minima of convex functions. Minimization of such functions can be used for approximation of finite sets and their clustering. We suggest to use the local discrete gradient (DG) method [Bag99] and the hybrid method between the cutting angle method and the discrete gradient method (DG+CAM) [BRZ05b] for the minimization of these functions. We report and analyze the results of numerical experiments.**Description:**2003004082

On global optimality conditions via separation functions

**Authors:**Rubinov, Alex , Uderzo, A.**Date:**2001**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 109, no. 2 (May 2001), p. 345-370**Full Text:**false**Reviewed:****Description:**The paper examines some axiomatic definitions of separation functions that can be employed fruitfully in the analysis of side-constrained extremum problems. A study of their general properties points out connections with abstract convex analysis and recent generalizations of Lagrangian approaches to duality and exact penalty methods. Many concrete examples are brought out.

An extended lifetime measure for telecommunication network

- Dzalilov, Zari, Ouveysi, Iradj, Rubinov, Alex

**Authors:**Dzalilov, Zari , Ouveysi, Iradj , Rubinov, Alex**Date:**2008**Type:**Text , Journal article**Relation:**Journal of Industrial and Management Optimization Vol. 4, no. 2 (2008), p. 329-337**Full Text:****Reviewed:****Description:**A new measure for network performance evaluation called topology lifetime was introduced in [4, 5]. This measure is based on the notion of unexpected traffic growth and can be used for comparison of topologies. We discuss some advantages and disadvantages of the approach of [4] and suggest some modifications to this approach. In particular we discuss how to evaluate the influence of a subgraph to the lifetime measure and introduce the notion of the order of a path. This notion is useful if we consider a possible extension to the set of working paths in order to support the traffic for the time that is needed for installation of new facilities.

**Authors:**Dzalilov, Zari , Ouveysi, Iradj , Rubinov, Alex**Date:**2008**Type:**Text , Journal article**Relation:**Journal of Industrial and Management Optimization Vol. 4, no. 2 (2008), p. 329-337**Full Text:****Reviewed:****Description:**A new measure for network performance evaluation called topology lifetime was introduced in [4, 5]. This measure is based on the notion of unexpected traffic growth and can be used for comparison of topologies. We discuss some advantages and disadvantages of the approach of [4] and suggest some modifications to this approach. In particular we discuss how to evaluate the influence of a subgraph to the lifetime measure and introduce the notion of the order of a path. This notion is useful if we consider a possible extension to the set of working paths in order to support the traffic for the time that is needed for installation of new facilities.

Best approximation in a class of normed spaces with star-shaped cone

- Mohebi, Hossein, Sadeghi, H., Rubinov, Alex

**Authors:**Mohebi, Hossein , Sadeghi, H. , Rubinov, Alex**Date:**2006**Type:**Text , Journal article**Relation:**Numerical Functional Analysis and Optimization Vol. 27, no. 3-4 (Apr-May 2006), p. 411-436**Full Text:**false**Reviewed:****Description:**We examine best approximation by closed sets in a class of normed spaces with star-shaped cones. It is assumed that the norm on the space X under consideration is generated by a star-shaped cone. First, we study best approximation by downward and upward sets, and then we use the results obtained as a tool for examination of best approximation by an arbitrary closed set.**Description:**C1**Description:**2003001837

A new algorithm for the placement of WLAN access point based on nonsmooth optimization technique

- Kouhbor, Shahnaz, Ugon, Julien, Kruger, Alexander, Rubinov, Alex, Branch, Philip

**Authors:**Kouhbor, Shahnaz , Ugon, Julien , Kruger, Alexander , Rubinov, Alex , Branch, Philip**Date:**2005**Type:**Text , Conference paper**Relation:**Paper presented at the 7th International Conference on Advanced Communication Technology, Phoenix Park, Korea : 21st February, 2005**Full Text:****Reviewed:****Description:**In wireless local area network (WLAN), signal coverage is obtained by proper placement of access points (APs). The impact of incorrect placement of APs is significant. If they are placed too far apart, they generate a coverage gap but if they are too close to each other, this leads to excessive co-channel interferences. In this paper, we describe a mathematical model we have developed to find the optimal number and location of APs. To solve the problem, we use an optimization algorithm developed at the University of Ballarat called discrete gradient algorithm. Results indicate that our model is able to solve optimal coverage problems for different numbers of users.**Description:**E1**Description:**2003001376

**Authors:**Kouhbor, Shahnaz , Ugon, Julien , Kruger, Alexander , Rubinov, Alex , Branch, Philip**Date:**2005**Type:**Text , Conference paper**Relation:**Paper presented at the 7th International Conference on Advanced Communication Technology, Phoenix Park, Korea : 21st February, 2005**Full Text:****Reviewed:****Description:**In wireless local area network (WLAN), signal coverage is obtained by proper placement of access points (APs). The impact of incorrect placement of APs is significant. If they are placed too far apart, they generate a coverage gap but if they are too close to each other, this leads to excessive co-channel interferences. In this paper, we describe a mathematical model we have developed to find the optimal number and location of APs. To solve the problem, we use an optimization algorithm developed at the University of Ballarat called discrete gradient algorithm. Results indicate that our model is able to solve optimal coverage problems for different numbers of users.**Description:**E1**Description:**2003001376

A feature selection approach for unsupervised classification based on clustering

- Rubinov, Alex, Soukhoroukova, Nadejda, Ugon, Julien

**Authors:**Rubinov, Alex , Soukhoroukova, Nadejda , Ugon, Julien**Date:**2004**Type:**Text , Conference paper**Relation:**Paper presented at Sixth International Conference on Optimization: Techniques and Applications (ICOTA) , University of Ballarat, Ballarat, Victoria : 9th-11th December 2004**Full Text:**false**Description:**Data have been collected for many years in different scientific (industrial, medical) research groups. Very often these groups kept all the the they could collect. It is possible that the data contains a lot of noisy features which do not bring any information, but make the problem more complicated. The additional study of eliminating non-informative and selecting informative features is very important in the area of Data Mining. There are several feature selection methods which were developed for supervised classification. The area of feature selection for unsupervised classification is not so developed. In this paper we present a new feature selection approach for unsupervised classification, based on clustering and nonsmooth optimisation techniques.**Description:**2003004085

An algorithm for monotonic global optimization problems

- Rubinov, Alex, Tuy, Hoang, Mays, Heather

**Authors:**Rubinov, Alex , Tuy, Hoang , Mays, Heather**Date:**2001**Type:**Text , Journal article**Relation:**Optimization Vol. 49, no. 3 (2001), p. 205-221**Full Text:**false**Reviewed:****Description:**We propose an algorithm to locate a global maximum of an increasing function subject to an increasing constraint on the cone of vectors with nonnegative coordinates. The algorithm is based on the outer approximation of the feasible set. We establish the convergence of the algorithm and provide a number of numerical experiments. We also discuss the types of constraints and objective functions for which the algorithm is best suited. © 2001 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint.

Downward sets and their separation and approximation properties

- Martinez-Legaz, Juan, Rubinov, Alex, Singer, Ivan

**Authors:**Martinez-Legaz, Juan , Rubinov, Alex , Singer, Ivan**Date:**2002**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 23, no. 2 (Jun 2002), p. 111-137**Full Text:**false**Reviewed:****Description:**We develop a theory of downward subsets of the space R-I, where I is a finite index set. Downward sets arise as the set of all solutions of a system of inequalities x is an element of R-I, f(t)(x) less than or equal to 0 (t is an element of T), where T is an arbitrary index set and each f(t) (t is an element of T) is an increasing function defined on R-I. These sets play an important role in some parts of mathematical economics and game theory. We examine some functions related to a downward set (the distance to this set and the plus-Minkowski gauge of this set, which we introduce here) and study lattices of closed downward sets and of corresponding distance functions. We discuss two kinds of duality for downward sets, based on multiplicative and additive min-type functions, respectively, and corresponding separation properties, and we give some characterizations of best approximations by downward sets. Some links between the multiplicative and additive cases are established.**Description:**2003000119

Monotonic analysis over cones : I

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

**Authors:**Dutta, J. , Martinez-Legaz, Juan , Rubinov, Alex**Date:**2004**Type:**Text , Journal article**Relation:**Optimization Vol. 53, no. 2 (2004), p. 129-146**Full Text:**false**Reviewed:****Description:**In this article, we study increasing and positively homogeneous functions defined on convex cones of locally convex spaces. This work is the first part in a series of studies to have a general view of the emerging area of Monotonic Analysis. We develop a general notion of so-called elementary functions, so that the generalized increasing and positively homogeneous functions can be represented as upper-envelopes of families of such functions. We also study many other associated properties like the description of support sets and normal and co-normal sets in a very general setting.**Description:**C1**Description:**2003000930

A global optimization approach to classification

- Bagirov, Adil, Rubinov, Alex, Yearwood, John

**Authors:**Bagirov, Adil , Rubinov, Alex , Yearwood, John**Date:**2002**Type:**Text , Journal article**Relation:**Optimization and Engineering Vol. 9, no. 7 (2002), p. 129-155**Full Text:**false**Reviewed:****Description:**In this paper is presented an hybrid algorithm for finding the absolute extreme point of a multimodal scalar function of many variables. The algorithm is suitable when the objective function is expensive to compute, the computation can be affected by noise and/or partial derivatives cannot be calculated. The method used is a genetic modification of a previous algorithm based on the Prices method. All information about behavior of objective function collected on previous iterates are used to chose new evaluation points. The genetic part of the algorithm is very effective to escape from local attractors of the algorithm and assures convergence in probability to the global optimum. The proposed algorithm has been tested on a large set of multimodal test problems outperforming both the modified Prices algorithm and classical genetic approach.**Description:**C1**Description:**2003000061

Optimality conditions in global optimization and their applications

**Authors:**Rubinov, Alex , Wu, Zhiyou**Date:**2009**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 120, no. 1 SPEC. ISS. (2009), p. 101-123**Full Text:**false**Reviewed:****Description:**In this paper we derive necessary and sufficient conditions for some problems of global minimization. Our approach is based on methods of abstract convexity: we use a representation of an upper semicontinuous function as the lower envelope of a family of convex functions. We discuss applications of conditions obtained to the examination of some tractable sufficient conditions for the global minimum and to the theory of inequalities. Â© 2007 Springer-Verlag.

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