A comparison of non-linear Lagrange and penalty functions for problems with a constraint
- Authors: Rubinov, Alex , Giri, Jason
- Date: 2001
- Type: Text , Conference paper
- Relation: Paper presented at 5th International Conference on Optimization: Techniques and Applications (ICOTA 2001), Hong Kong : 15th -17th December, 2001 p. 360-366
- Full Text: false
- Reviewed:
- Description: E1
- Description: 2003003929
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
- Full Text:
- 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
An algorithm for monotonic global optimization problems
- 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.
Difference inclusions with delay of economic growth
- Authors: Dzalilov, Zari , Ivanov, Anatoli , Rubinov, Alex
- Date: 2001
- Type: Text , Journal article
- Relation: Dynamic Systems and Applications Vol. 10 , no. (2001), p. 283-293
- Full Text: false
- Reviewed:
- Description: A difference inclusion wit.h delay is proposed as a modified model of maeroe<.'onomical growth. The classical assumpt.ion of t.he homogeneity of the nonlinear feedback involved in the model implies t.he existence of a ray of equilibria. Any dynamics in the model is shown to be convergent. t.o either an equilibrium on the ray or to the zero equilibrium.
- Description: C1
- Description: 2003002561
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
- 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.
Equilibrium with fixed budgets and superlinear connections
- Authors: Rubinov, Alex , Glover, Barney
- Date: 2001
- Type: Text , Journal article
- Relation: ANZIAM Journal Vol. 42, no. 4 (2001), p. 462-480
- Full Text: false
- Reviewed:
- Description: We study models of economic equilibrium with fixed budgets and assuming superlinear connections between consumption and production. Extremal problems and the existence of equilibria are discussed for such models along with some related differential properties. Examples to illustrate the broad nature of the model are discussed. © Australian Mathematical Society 2001.
Extended Lagrange and penalty functions in optimization
- Authors: Rubinov, Alex , Yang, Xiao , Glover, Barney
- Date: 2001
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 111, no. 2 (Nov 2001), p. 381-405
- Full Text: false
- Reviewed:
- Description: We consider nonlinear Lagrange and penalty functions for optimization problems with a single constraint. The convolution of the objective function and the constraint is accomplished by an increasing positively homogeneous of the first degree function. We study necessary and also sufficient conditions for the validity of the zero duality gap property for both Lagrange and penalty functions and for the exact penalization. We also study the so-called regular weak separation functions.
General lagrange-type functions in constrained global optimization part I : Auxiliary functions and optimality conditions
- Authors: Evtushenko, Yu G. , Rubinov, Alex , Zhadan, V. G.
- Date: 2001
- Type: Text , Journal article
- Relation: Optimization Methods and Software Vol. 16, no. 1-4 (2001), p. 193-230
- Full Text: false
- Reviewed:
- Description: The paper contains some new results and a survey of some known results related to auxiliary (Lagrange-type) functions in constrained optimization. We show that auxiliary functions can be constructed by means of two-step convolution of constraints and the objective function and present some conditions providing the validity of the zero duality gap property. We show that auxiliary functions are closely related to the so-called separation functions in the image space of the constrained problem under consideration. The second part of the paper (see Evtushenko et al., General Lagrange-type functions in constrained global optimization. Part II: Exact Auxiliary functions. Optimization Methods and Software) contains results related to exact auxiliary functions. © 2001 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint, a member of the Taylor & Francis Group.
General lagrange-type functions in constrained global optimization part II : Exact auxiliary functions
- Authors: Evtushenko, Yu G. , Rubinov, Alex , Zhadan, V. G.
- Date: 2001
- Type: Text , Journal article
- Relation: Optimization Methods and Software Vol. 16, no. 1-4 (2001), p. 231-256
- Full Text: false
- Reviewed:
- Description: This paper is a continuation of [13]. For each constrained optimization problem we consider certain unconstrained problems, which are constructed by means of auxiliary (Lagrange-type) functions. We study only exact auxiliary functions, it means that the set of their global minimizers coincides with the solution set of the primal constrained optimization problem. Sufficient conditions for the exactness of an auxiliary function are given. These conditions are obtained without assumption that the Lagrange function has a saddle point. Some examples of exact auxiliary functions are given. © 2001 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint, a member of the Taylor & Francis Group.
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.
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.
Topical and sub-topical functions, downward sets and abstract convexity
- Authors: Rubinov, Alex , Singer, Ivan
- Date: 2001
- Type: Text , Journal article
- Relation: Optimization Vol. 50, no. 5-6 (2001), p. 307-351
- Full Text: false
- Reviewed:
- Description: We study topical and sub-topical functions (i.e., functions f : Rn → R = [-∞, +∞] which are increasing in the natural partial ordering of Rn and additively homogeneous, respectively additively sub-homogeneous), and downward sets (i.e., subsets of ℝn which contain, along with each element, all smaller elements), in the framework of abstract convex analysis, with the aid of the additive min-type coupling function
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
- Reviewed:
- Description: C1
- Description: 2003002564
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
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 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
Abstract convexity and non-smooth analysis : Global aspects
- Authors: Ioffe, A. D , Rubinov, Alex
- Date: 2002
- Type: Text , Journal article
- Relation: Advances in Mathematical Economics Vol. 4, no. (2002), p. 1-23
- Full Text: false
- Reviewed:
- Description: We study some types of convergence of abstract convex functions. We show that under certain conditions, convergence of support sets of a sequence of abstract convex functions implies graph convergence of the support sets at a point and the global subdifferential. Some examples and counterexamples are given.
- Description: C1
- Description: 2003000146
Abstract convexity of positive homogenous functions
- Authors: Rubinov, Alex , Dzalilov, Zari
- Date: 2002
- Type: Text , Journal article
- Relation: Journal of Statistics and Management Systems Vol. 5, no. 1 (2002), p. 1-19
- Full Text:
- Reviewed:
- Description: This paper is a survey of recent results to abstract convexity of positively homogeneous functions, that is, to a representation of such functions as the upper (lower) envelope of sets of fairly simple functions. In particular, we study sup-min and max-min representations of positively homogeneous of degree one functions through linear functions and various kinds of the so-called exhausters.
- Description: C1
- Description: 2003003353
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
- 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
Downward sets and their separation and approximation properties
- 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