B-convex sets and functions
- Authors: Adilov, G. , Rubinov, Alex
- Date: 2006
- Type: Text , Journal article
- Relation: Numerical Functional Analysis and Optimization Vol. 27, no. 3-4 (Apr-May 2006), p. 237-257
- Full Text: false
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- Description: A subset B of R-+(n) is B-convex if for all x, y is an element of B and all t is an element of [0, 1] one has max (tx, y) is an element of B. These sets were first investigated in [1, 2]. In this paper, we examine radiant B-convex sets and also introduce and study B-convex functions.
- Description: C1
- Description: 2003001836
Global optimization of marginal functions with applications to economic equilibrium
- 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.
A global optimization approach to classification
- 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
A global optimisation approach to classification in medical diagnosis and prognosis
- Authors: Bagirov, Adil , Rubinov, Alex , Yearwood, John , Stranieri, Andrew
- Date: 2001
- Type: Text , Conference paper
- Relation: Paper presented at 34th Hawaii International Conference on System Sciences, HICSS-34, Maui, Hawaii, USA : 3rd-6th January 2001
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- Description: In this paper global optimisation-based techniques are studied in order to increase the accuracy of medical diagnosis and prognosis with FNA image data from the Wisconsin Diagnostic and Prognostic Breast Cancer databases. First we discuss the problem of determining the most informative features for the classification of cancerous cases in the databases under consideration. Then we apply a technique based on convex and global optimisation to breast cancer diagnosis. It allows the classification of benign cases and malignant ones and the subsequent diagnosis of patients with very high accuracy. The third application of this technique is a method that calculates centres of clusters to predict when breast cancer is likely to recur in patients for which cancer has been removed. The technique achieves higher accuracy with these databases than reported elsewhere in the literature.
- Description: 2003003950
A multidimensional descent method for global optimization
- Authors: Bagirov, Adil , Rubinov, Alex , Zhang, Jiapu
- Date: 2009
- Type: Text , Journal article
- Relation: Optimization Vol. 58, no. 5 (2009), p. 611-625
- Full Text: false
- Reviewed:
- Description: This article presents a new multidimensional descent method for solving global optimization problems with box-constraints. This is a hybrid method where local search method is used for a local descent and global search is used for further multidimensional search on the subsets of intersection of cones generated by the local search method and the feasible region. The discrete gradient method is used for local search and the cutting angle method is used for global search. Two-and three-dimensional cones are used for the global search. Such an approach allows one, as a rule, to escape local minimizers which are not global ones. The proposed method is local optimization method with strong global search properties. We present results of numerical experiments using both smooth and non-smooth global optimization test problems. These results demonstrate that the proposed algorithm allows one to find a global or a near global minimizer.
Cutting angle method and a local search
- Authors: Bagirov, Adil , Rubinov, Alex
- Date: 2003
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 27, no. 2-3 (Nov 2003), p. 193-213
- Full Text: false
- Reviewed:
- Description: The paper deals with combinations of the cutting angle method in global optimization and a local search. We propose to use special transformed objective functions for each intermediate use of the cutting angle method. We report results of numerical experiments which demonstrate that the proposed approach is very beneficial in the search for a global minimum.
- Description: C1
- Description: 2003000438
Penalty functions with a small penalty parameter : Numerical experiments
- Authors: Bagirov, Adil , Rubinov, Alex
- Date: 2003
- Type: Text , Conference paper
- Relation: Paper presented at Industrial Optimization Conference 2003, Perth : 30th September, 2002
- Full Text: false
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- Description: E1
- Description: 2003000432
An algorithm for clustering based on non-smooth optimization techniques
- Authors: Bagirov, Adil , Rubinov, Alex , Sukhorukova, Nadezda , Yearwood, John
- Date: 2003
- Type: Text , Journal article
- Relation: International Transactions in Operational Research Vol. 10, no. 6 (2003), p. 611-617
- Full Text: false
- Reviewed:
- Description: The problem of cluster analysis is formulated as a problem of non-smooth, non-convex optimization, and an algorithm for solving the cluster analysis problem based on non-smooth optimization techniques is developed. We discuss applications of this algorithm in large databases. Results of numerical experiments are presented to demonstrate the effectiveness of this algorithm.
- Description: C1
- Description: 2003000422
Using global optimization to improve classification for medical diagnosis and prognosis
- Authors: Bagirov, Adil , Rubinov, Alex , Yearwood, John
- Date: 2001
- Type: Text , Journal article
- Relation: Topics in health information management Vol. 22, no. 1 (2001), p. 65-74
- Full Text: false
- Description: Global optimization-based techniques are studied in order to increase the accuracy of medical diagnosis and prognosis with data from various databases. First, we discuss feature selection, the problem of determining the most informative features for classification in the databases under consideration. Then, we apply a technique based on convex and global optimization for classification in these databases. The third application of this technique is a method that calculates centers of clusters to predict when breast cancer is likely to recur in patients for which cancer has been removed. The technique achieves high accuracy with these databases. Better classifiers will lead to improved assistance in making medical diagnostic and prognostic decisions.
- Description: 2003003662
Unsupervised and supervised data classification via nonsmooth and global optimisation
- Authors: Bagirov, Adil , Rubinov, Alex , Sukhorukova, Nadezda , Yearwood, John
- Date: 2003
- Type: Text , Journal article
- Relation: Top Vol. 11, no. 1 (2003), p. 1-92
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- Reviewed:
- Description: We examine various methods for data clustering and data classification that are based on the minimization of the so-called cluster function and its modications. These functions are nonsmooth and nonconvex. We use Discrete Gradient methods for their local minimization. We consider also a combination of this method with the cutting angle method for global minimization. We present and discuss results of numerical experiments.
- Description: C1
- Description: 2003000421
Local optimization method with global multidimensional search
- Authors: Bagirov, Adil , Rubinov, Alex , Zhang, Jiapu
- Date: 2005
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 32, no. 2 (2005), p. 161-179
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- Reviewed:
- Description: This paper presents a new method for solving global optimization problems. We use a local technique based on the notion of discrete gradients for finding a cone of descent directions and then we use a global cutting angle algorithm for finding global minimum within the intersection of the cone and the feasible region. We present results of numerical experiments with well-known test problems and with the so-called cluster function. These results confirm that the proposed algorithms allows one to find a global minimizer or at least a deep local minimizer of a function with a huge amount of shallow local minima. © Springer 2005.
- Description: C1
- Description: 2003001351
Conical decomposition and vector lattices with respect to several preorders
- Authors: Baratov, Rishat , Rubinov, Alex
- Date: 2006
- Type: Text , Journal article
- Relation: Taiwanese Journal of Mathematics Vol. 10, no. 2 (2006), p. 265-298
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- 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
Abstract convexity and augmented Lagrangians
- Authors: Burachik, Regina , Rubinov, Alex
- Date: 2007
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 18, no. 2 (2007), p. 413-436
- Full Text: false
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- Description: The ultimate goal of this paper is to demonstrate that abstract convexity provides a natural language and a suitable framework for the examination of zero duality gap properties and exact multipliers of augmented Lagrangians. We study augmented Lagrangians in a very general setting and formulate the main definitions and facts describing the augmented Lagrangian theory in terms of abstract convexity tools. We illustrate our duality scheme with an application to stochastic semiinfinite optimization. © 2007 Society for Industrial and Applied Mathematics.
- Description: C1
- Description: 2003005362
On abstract convexity and set valued analysis
- Authors: Burachik, Regina , Rubinov, Alex
- Date: 2008
- Type: Text , Journal article
- Relation: Journal of Nonlinear and Convex Analysis Vol. 9, no. 1 (2008), p. 105-123
- Full Text: false
- Reviewed:
- Description: Given a set L subset of R-X of functions defined on X, we consider abstract monotone (or, for short, L-monotone) multivalued operators T : X paired right arrows L. We extend the definition of enlargement of monotone operators to this framework and study semicontinuity properties of these mappings. We prove that sequential outer semicontinuity, which holds for maximal monotone operators and their enlargements in the classical case (i.e., when L = X* and X is a Banach space), holds also in our abstract setting. We also show through examples that some properties, known to hold in the classical case, may no longer be valid in the abstract setting. One of these properties is the maximality of the subdifferential and another one is the lack of inner semicontinuity of (point-to-set) monotone operators in the interior of their domain. We also focus on the structure of both the abstract subdifferential and the abstract epsilon-subdifferential. This is a key question in abstract convexity because these sets may be very large for certain choices of L and therefore it is important to be able to represent them by means of some special elements of the set of "affine" functions induced by L.
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
- Full Text: false
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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
A method of truncated codifferential with application to some problems of cluster analysis
- 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
Monotonic analysis over cones : I
- 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
Monotonic analysis over cones : II
- Authors: Dutta, J. , Martinez-Legaz, Juan , Rubinov, Alex
- Date: 2004
- Type: Text , Journal article
- Relation: Optimization Vol. 53, no. 5-6 (2004), p. 529-547
- Full Text: false
- Reviewed:
- Description: In this article, we study the class of increasing and convex along rays (ICAR) functions over a cone. Apart from studying its basic properties, we study them from the point of view of Abstract Convexity. Further, we study the relation between the ICAR and Lipschitz functions and the properties under which an ICAR function has a Lipschitz behaviour. We also study the class of decreasing and convex along rays functions (DCAR).
- Description: C1
- Description: 2003000931
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
- Full Text: false
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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
An extended lifetime measure for telecommunication network
- 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
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- 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.