Chebyshev multivariate polynomial approximation : alternance interpretation
- Authors: Sukhorukova, Nadezda , Ugon, Julien , Yost, David
- Date: 2018
- Type: Text , Book chapter
- Relation: 2016 Matrix Annals p. 177-182
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- Description: In this paper, we derive optimality conditions for Chebyshev approximation of multivariate functions. The theory of Chebyshev (uniform) approximation for univariate functions was developed in the late nineteenth and twentieth century. The optimality conditions are based on the notion of alternance (maximal deviation points with alternating deviation signs). It is not clear, however, how to extend the notion of alternance to the case of multivariate functions. There have been several attempts to extend the theory of Chebyshev approximation to the case of multivariate functions. We propose an alternative approach, which is based on the notion of convexity and nonsmooth analysis.
Minimization of the sum of minima of convex functions and its application to clustering
- Authors: Rubinov, Alex , Soukhoroukova, Nadejda , Ugon, Julien
- Date: 2005
- Type: Text , Book chapter
- Relation: Continuous Optimization Chapter p. 409-434
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- Description: We study functions that can be represented as the sum of minima of convex functions. Minimization of such functions can be used for approximation of finite sets and their clustering. We suggest to use the local discrete gradient (DG) method [Bag99] and the hybrid method between the cutting angle method and the discrete gradient method (DG+CAM) [BRZ05b] for the minimization of these functions. We report and analyze the results of numerical experiments.
- Description: 2003004082
Supervised data classification via max-min separability
- Authors: Ugon, Julien , Bagirov, Adil
- Date: 2005
- Type: Text , Book chapter
- Relation: Continuous Optimization: Current Trends and Modern Applications Chapter p. 175-208
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- Description: B1
- Description: 2003001268