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The nonlinear and augmented Lagrangians for nonconvex optimization problems with a single constraint

- Rubinov, Alex, Gasimov, Rafail

  • Authors: Rubinov, Alex , Gasimov, Rafail
  • Date: 2002
  • Type: Text , Journal article
  • Relation: Applied and Computational Mathematics Vol. 1, no. 2 (2002), p. 142-157
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  • Description: The paper contains the survey of some recent results obtained by the authors and their colleagues. We study zero duality gap properties for optimization problems with a single constraint with respect to a nonlinear penalization. The penalty function is constructed as a convolution of the objective function and the constraint by means of IPH (increasing positively homogeneous) functions. The main results are obtained for penalization by strictly IPH functions. We also examine augmented Lagrangians for optimization problems with a single constraint. We establish some links between augmented Lagrangians and Lagrange-type functions and propose a new kind of Lagrange-type functions for the problems with a single inequality constraint.
  • Description: C1
  • Description: 2003000115

The zero duality gap property and lower semicontinuity of the perturbation function

- Rubinov, Alex, Huang, X. X., Yang, Xiao

  • Authors: Rubinov, Alex , Huang, X. X. , Yang, Xiao
  • Date: 2002
  • Type: Text , Journal article
  • Relation: Mathematics of Operations Research Vol. 27, no. 4 (2002), p. 775-791
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  • Description: We examine the validity of the zero duality gap properties for two important dual schemes: a generalized augmented Lagrangian dual scheme and a nonlinear Lagrange-type dual scheme. The necessary and sufficient conditions for the zero duality gap property to hold are established in terms of the lower semicontinuity of the perturbation functions.
  • Description: 2003000117

Two porosity results in monotonic analysis

- Rubinov, Alex, Zaslavski, Alexander

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
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  • 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

- Dzalilov, Zari, Ivanov, Anatoli, Rubinov, Alex

  • 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
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  • 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

- Vladimirov, Alexander, Rubinov, Alex

  • Authors: Vladimirov, Alexander , Rubinov, Alex
  • Date: 2001
  • Type: Text , Journal article
  • Relation: Journal of Convex Analysis Vol. 8, no. 2 (2001), p. 387-399
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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.

Equilibrium with fixed budgets and superlinear connections

- Rubinov, Alex, Glover, Barney

  • Authors: Rubinov, Alex , Glover, Barney
  • Date: 2001
  • Type: Text , Journal article
  • Relation: ANZIAM Journal Vol. 42, no. 4 (2001), p. 462-480
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  • 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

- Rubinov, Alex, Yang, Xiao, Glover, Barney

  • 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
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  • 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

- Evtushenko, Yu G., Rubinov, Alex, Zhadan, V. G.

  • 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
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  • 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

- Evtushenko, Yu G., Rubinov, Alex, Zhadan, V. G.

  • 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
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  • 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

- 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
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  • 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

- Rubinov, Alex, Uderzo, A.

  • 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
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  • 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

- Rubinov, Alex, Singer, Ivan

  • Authors: Rubinov, Alex , Singer, Ivan
  • Date: 2001
  • Type: Text , Journal article
  • Relation: Optimization Vol. 50, no. 5-6 (2001), p. 307-351
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  • 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

- Ivanov, Anatoli, Dzalilov, Zari, Rubinov, Alex

Using global optimization to improve classification for medical diagnosis and prognosis

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

  • 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
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  • 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