Vector optimization problems with nonconvex preferences
- Authors: Huang, N. J. , Rubinov, Alex , Yang, Xiao
- Date: 2008
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 40, no. 4 (2008), p. 765-777
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- Description: In this paper, some vector optimization problems are considered where pseudo-ordering relations are determined by nonconvex cones in Banach spaces. We give some characterizations of solution sets for vector complementarity problems and vector variational inequalities. When the nonconvex cone is the union of some convex cones, it is shown that the solution set of these problems is either an intersection or an union of the solution sets of all subproblems corresponding to each of these convex cones depending on whether these problems are defined by the nonconvex cone itself or its complement. Moreover, some relations of vector complementarity problems, vector variational inequalities, and minimal element problems are also given. © 2007 Springer Science+Business Media, Inc.
- Description: C1
A lagrange penalty reformulation method for constrained optimization
- Authors: Rubinov, Alex , Yang, Xiao , Zhou, Y. Y.
- Date: 2007
- Type: Text , Journal article
- Relation: Optimization Letters Vol. 1, no. 2 (2007), p. 145-154
- Full Text: false
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- Description: In this paper a constrained optimization problem is transformed into an equivalent one in terms of an auxiliary penalty function. A Lagrange function method is then applied to this transformed problem. Zero duality gap and exact penalty results are obtained without any coercivity assumption on either the objective function or constraint functions. © 2006 Springer-Verlag.
- Description: C1
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
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
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- 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
Scalarization and nonlinear scalar duality for vector optimization with preferences that are not necessarily a pre-order relation
- Authors: Rubinov, Alex , Gasimov, Rafail
- Date: 2004
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 29, no. 4 (2004), p. 455-477
- Full Text: false
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- Description: We consider problems of vector optimization with preferences that are not necessarily a pre-order relation. We introduce the class of functions which can serve for a scalarization of these problems and consider a scalar duality based on recently developed methods for non-linear penalization scalar problems with a single constraint.
- Description: C1
- Description: 2003000932
Penalty functions with a small penalty parameter
- Authors: Rubinov, Alex , Yang, Xiao , Bagirov, Adil
- Date: 2002
- Type: Text , Journal article
- Relation: Optimization Methods and Software Vol. 17, no. 5 (2002), p. 931-964
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- Description: In this article, we study the nonlinear penalization of a constrained optimization problem and show that the least exact penalty parameter of an equivalent parametric optimization problem can be diminished. We apply the theory of increasing positively homogeneous (IPH) functions so as to derive a simple formula for computing the least exact penalty parameter for the classical penalty function through perturbation function. We establish that various equivalent parametric reformulations of constrained optimization problems lead to reduction of exact penalty parameters. To construct a Lipschitz penalty function with a small exact penalty parameter for a Lipschitz programming problem, we make a transformation to the objective function by virtue of an increasing concave function. We present results of numerical experiments, which demonstrate that the Lipschitz penalty function with a small penalty parameter is more suitable for solving some nonconvex constrained problems than the classical penalty function.
- Description: 2003000116
The zero duality gap property and lower semicontinuity of the perturbation function
- 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
- Full Text: false
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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
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.