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
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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
- 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
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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
- 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
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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.
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
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- 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 for minimization of quasidifferentiable functions
- Authors: Bagirov, Adil
- Date: 2002
- Type: Text , Journal article
- Relation: Optimization Methods and Software Vol. 17, no. 1 (2002), p. 31-60
- Full Text: false
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- Description: In this paper, we propose a new method for the unconstrained minimization of a function presented as a difference of two convex functions. This method is based on continuous approximations to the Demyanov-Rubinov quasidifferential. First, a terminating algorithm for the computation of a descent direction of the objective function is described. Then we present a minimization algorithm and study its convergence. An implementable version of this algorithm is discussed. Finally, we report the results of preliminary numerical experiments.
- Description: C1
- Description: 2003000064
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
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- 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
The nonlinear and augmented Lagrangians for nonconvex optimization problems with a single constraint
- Authors: Rubinov, Alex , Gasimov, Rafail
- Date: 2002
- Type: Text , Journal article
- Relation: Applied and Computational Mathematics Vol. 1, no. 2 (2002), p. 142-157
- Full Text: false
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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
Comparative analysis of the cutting angle and simulated annealing methods in global optimization
- Authors: Bagirov, Adil , Zhang, Jiapu
- Date: 2003
- Type: Text , Journal article
- Relation: Optimization Vol. 52, no. 4-5 (2003), p. 363-378
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- Description: This article presents a comparative analysis of two methods of global optimization: the simulated annealing method and a method based on a combination of the cutting angle method and a local search. This analysis is carried out using results of numerical experiments. These results demonstrate that the combined method is more effective than the simulated annealing method.
- Description: C1
- Description: 2003000436
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
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
On augmented lagrangians for optimization problems with a single constraint
- 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
Weak stationarity : Eliminating the gap between necessary and sufficient conditions
- Authors: Kruger, Alexander
- Date: 2004
- Type: Text , Journal article
- Relation: Optimization Vol. 53, no. 2 (Apr 2004), p. 147-164
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- Description: Starting from known necessary extremality conditions in terms of strict subdifferentials and normals the notion of weak stationarity is introduced. It is defined in terms of initial space elements. The necessary conditions become necessary and sufficient (for stationarity).
- Description: 2003000887
A dual condition for the convex subdifferential sum formula with applications
- Authors: Burachik, Regina , Jeyakumar, Vaithilingam
- Date: 2005
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 12, no. 2 (2005), p. 279-290
- Full Text: false
- Reviewed:
- Description: C1
- Description: 2003002555
H-infinity via a nonsmooth, nonconvex optimization approach
- Authors: Mammadov, Musa , Orsi, Robert
- Date: 2005
- Type: Text , Journal article
- Relation: Pacific Journal of Optimization Vol. 1, no. 2 (2005), p. 405-420
- Full Text: false
- Reviewed:
- Description: A numerical method for solving the H-infinity synthesis problem is presented. The problem is posed as an unconstrained, nonsmooth, nonconvex minimization problem. The optimization variables consist solely of the entries of the output feedback matrix. No additional variables, such as Lyapunov variables, need to be introduced. The main part of the optimization procedure uses a line search mechanism where the descent direction is defined by a recently introduced dynamical systems approach. Numerical results for various benchmark problems are included in the paper. In addition, the effectiveness of a preliminary part of the algorithm for successfully and quickly finding stabilizing controllers is also demonstrated.
- Description: C1
- Description: 2003001382
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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- 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
Max-min separability
- Authors: Bagirov, Adil
- Date: 2005
- Type: Text , Journal article
- Relation: Optimization Methods and Software Vol. 20, no. 2-3 (2005), p. 271-290
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- Description: We consider the problem of discriminating two finite point sets in the n-dimensional space by a finite number of hyperplanes generating a piecewise linear function. If the intersection of these sets is empty, then they can be strictly separated by a max-min of linear functions. An error function is introduced. This function is nonconvex piecewise linear. We discuss an algorithm for its minimization. The results of numerical experiments using some real-world datasets are presented, which show the effectiveness of the proposed approach.
- Description: C1
- Description: 2003001350
On the absence of duality gap for Lagrange-type functions
- 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:
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- 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
Optimization approach for clustering datasets with weights
- Authors: Ghosh, Ranadhir , Rubinov, Alex , Zhang, Jiapu
- Date: 2005
- Type: Text , Journal article
- Relation: Optimization Methods & Software Vol. 20, no. 2-3 (Apr-Jun 2005), p. 329-345
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- Description: We introduce datasets with weights and suggest using the minimization of some highly nonsmooth functions for clustering of such datasets. Datasets with weights often appear as the result of an approximation of large-scale datasets. We examine such approximations and also consider the application of datasets with weights to examine self-organizing maps. Results of some numerical experiments are presented and discussed.
- Description: C1
- Description: 2003001366
Separation in B-convexity
- Authors: Rubinov, Alex , Briec, W. , Horvath, C. D
- Date: 2005
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 1, no. 1 (2005), p. 13-30
- Full Text: false
- Reviewed:
- Description: A subset B of R
- Description: C1
- Description: 2003001426