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
- Full Text:
- 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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- Reviewed:
- Description: B1
- Description: 2003001268
Piecewise linear classifiers based on nonsmooth optimization approaches
- Authors: Bagirov, Adil , Kasimbeyli, Refail , Ozturk, Gurkan , Ugon, Julien
- Date: 2014
- Type: Text , Book chapter
- Relation: Optimization in Science and Engineering p. 1-32
- Full Text: false
- Reviewed:
- Description: Nonsmooth optimization provides efficient algorithms for solving many machine learning problems. In particular, nonsmooth optimization approaches to supervised data classification problems lead to the design of very efficient algorithms for their solution. In this chapter, we demonstrate how nonsmooth optimization algorithms can be applied to design efficient piecewise linear classifiers for supervised data classification problems. Such classifiers are developed using a max–min and a polyhedral conic separabilities as well as an incremental approach. We report results of numerical experiments and compare the piecewise linear classifiers with a number of other mainstream classifiers.
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
- Full Text:
- Reviewed:
- 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.
Schur functions for approximation problems
- Authors: Sukhorukova, Nadezda , Ugon, Julien , Yost, David
- Date: 2020
- Type: Text , Book chapter
- Relation: 2018 Matrix Annals p. 331-337
- Full Text: false
- Reviewed:
- Description: In this paper we propose a new approach to least squares approximation problems. This approach is based on partitioning and Schur function. The nature of this approach is combinatorial, while most existing approaches are based on algebra and algebraic geometry. This problem has several practical applications. One of them is curve clustering. We use this application to illustrate the results.