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20102 Applied Mathematics
2Downward sets
10101 Pure Mathematics
1Best approximation
1Down and upward and upward sets
1Function evaluation
1Global minium
1Global optimisation
1Metric projection
1Necessary and sufficient conditions
1Numerical methods
1Operations research
1Proximinal sets
1Relatively downward
1Relatively upward sets
1Star-shaped set
1Upward sets

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Best approximation by downward sets with applications

- Rubinov, Alex, Mohebi, Hossein

**Authors:**Rubinov, Alex , Mohebi, Hossein**Date:**2006**Type:**Text , Journal article**Relation:**Analysis in Theory and Applications Vol. 22, no. 1 (2006), p. 20-40**Full Text:****Reviewed:****Description:**We develop a theory of downward sets for a class of normed ordered spaces. We study best approximation in a normed ordered space X by elements of downward sets, and give necessary and sufficient conditions for any element of best approximation by a closed downward subset of X. We also characterize strictly downward subsets of X, and prove that a downward subset of X is strictly downward if and only if each its boundary point is Chebyshev. The results obtained are used for examination of some Chebyshev pairs (W,x), where x E X and W is a closed downward subset of X.**Description:**C1**Description:**2003001535

**Authors:**Rubinov, Alex , Mohebi, Hossein**Date:**2006**Type:**Text , Journal article**Relation:**Analysis in Theory and Applications Vol. 22, no. 1 (2006), p. 20-40**Full Text:****Reviewed:****Description:**We develop a theory of downward sets for a class of normed ordered spaces. We study best approximation in a normed ordered space X by elements of downward sets, and give necessary and sufficient conditions for any element of best approximation by a closed downward subset of X. We also characterize strictly downward subsets of X, and prove that a downward subset of X is strictly downward if and only if each its boundary point is Chebyshev. The results obtained are used for examination of some Chebyshev pairs (W,x), where x E X and W is a closed downward subset of X.**Description:**C1**Description:**2003001535

Best approximation in a class of normed spaces with star-shaped cone

- Mohebi, Hossein, Sadeghi, H., Rubinov, Alex

**Authors:**Mohebi, Hossein , Sadeghi, H. , Rubinov, Alex**Date:**2006**Type:**Text , Journal article**Relation:**Numerical Functional Analysis and Optimization Vol. 27, no. 3-4 (Apr-May 2006), p. 411-436**Full Text:**false**Reviewed:****Description:**We examine best approximation by closed sets in a class of normed spaces with star-shaped cones. It is assumed that the norm on the space X under consideration is generated by a star-shaped cone. First, we study best approximation by downward and upward sets, and then we use the results obtained as a tool for examination of best approximation by an arbitrary closed set.**Description:**C1**Description:**2003001837

Metric projection onto a closed set : Necessary and sufficient conditions for the global minimum

- Mohebi, Hossein, Rubinov, Alex

**Authors:**Mohebi, Hossein , Rubinov, Alex**Date:**2006**Type:**Text , Journal article**Relation:**Mathematics of Operations Research Vol. 31, no. 1 (2006), p. 124-132**Full Text:**false**Reviewed:****Description:**Necessary and sufficient conditions for a local minimum form a well-developed chapter of optimization theory. Determination of such conditions for the global minimum is a challenging problem. Useful conditions are currently known only for a few classes of nonconvex optimization problems. It is important to find different classes of problems for which the required conditions can be obtained. In this paper we examine one of these classes: the minimization of the distance to an arbitrary closed set in a class of ordered normed spaces. We use the structure of the objective function in order to present necessary and sufficient conditions that give a clear understanding of the structure of a global minimizer and can be easily verified for some problems under consideration. © 2006 INFORMS.**Description:**C1**Description:**2003001835

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