An algorithm for clustering based on non-smooth optimization techniques
- Authors: Bagirov, Adil , Rubinov, Alex , Sukhorukova, Nadezda , Yearwood, John
- Date: 2003
- Type: Text , Journal article
- Relation: International Transactions in Operational Research Vol. 10, no. 6 (2003), p. 611-617
- Full Text: false
- Reviewed:
- Description: The problem of cluster analysis is formulated as a problem of non-smooth, non-convex optimization, and an algorithm for solving the cluster analysis problem based on non-smooth optimization techniques is developed. We discuss applications of this algorithm in large databases. Results of numerical experiments are presented to demonstrate the effectiveness of this algorithm.
- Description: C1
- Description: 2003000422
Finite alternation theorems and a constructive approach to piecewise polynomial approximation in chebyshev norm
- Authors: Crouzeix, Jean-Pierre , Sukhorukova, Nadezda , Ugon, Julien
- Date: 2020
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 123-147. http://purl.org/au-research/grants/arc/DP180100602
- Full Text: false
- Reviewed:
- Description: One of the purposes in this paper is to provide a better understanding of the alternance property which occurs in Chebyshev polynomial approximation and continuous piecewise polynomial approximation problems. In the first part of this paper, we prove that alternating sequences of any continuous function are finite in any given segment and then propose an original approach to obtain new proofs of the well known necessary and sufficient optimality conditions. There are two main advantages of this approach. First of all, the proofs are intuitive and easy to understand. Second, these proofs are constructive and therefore they lead to new alternation-based algorithms. In the second part of this paper, we develop new local optimality conditions for free knot polynomial spline approximation. The proofs for free knot approximation are relying on the techniques developed in the first part of this paper. The piecewise polynomials are required to be continuous on the approximation segment. © 2020, Springer Nature B.V.
Asymmetrical correlation test for constructing Super Bayesian Influence Networks for financial intermarket influence analysis
- Authors: Pan, Heping , Sukhorukova, Nadezda
- Date: 2004
- Type: Text , Conference paper
- Relation: Paper presented at CIMCA 2004: International Conference on Computational Intelligence for Modelling, Control & Automation, Gold Coast, Queensland : 12th July, 2004
- Full Text: false
- Reviewed:
- Description: E1
- Description: 2003000913
Generalised rational approximation and its application to improve deep learning classifiers
- Authors: Peiris, V , Sharon, Nir , Sukhorukova, Nadezda , Ugon, Julien
- Date: 2021
- Type: Text , Journal article
- Relation: Applied Mathematics and Computation Vol. 389, no. (2021), p.
- Relation: https://purl.org/au-research/grants/arc/DP180100602
- Full Text: false
- Reviewed:
- Description: A rational approximation (that is, approximation by a ratio of two polynomials) is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non-Lipschitz functions, where polynomial approximations are not efficient. We prove that the optimisation problems appearing in the best uniform rational approximation and its generalisation to a ratio of linear combinations of basis functions are quasiconvex even when the basis functions are not restricted to monomials. Then we show how this fact can be used in the development of computational methods. This paper presents a theoretical study of the arising optimisation problems and provides results of several numerical experiments. We apply our approximation as a preprocessing step to deep learning classifiers and demonstrate that the classification accuracy is significantly improved compared to the classification of the raw signals. © 2020
- Description: This research was supported by the Australian Research Council (ARC), Solving hard Chebyshev approximation problems through nonsmooth analysis (Discovery Project DP180100602 ). This research was partially sponsored by Tel Aviv-Swinburne Research Collaboration Grant (2019).
Analysis of the Australian credit database
- Authors: Rubinov, Alex , Sukhorukova, Nadezda , Yearwood, John
- Date: 2003
- Type: Text , Conference paper
- Relation: Paper presented at the Symposium on Industrial Optimisation and the 9th Australian Optimisation Day, Perth : 30th September, 2002
- Full Text: false
- Reviewed:
- Description: E1
- Description: 2003000353
A generalization of the Remez algorithm to a class of linear spline approximation problems with constraints on spline parameters
- Authors: Sukhorukova, Nadezda
- Date: 2008
- Type: Text , Journal article
- Relation: Optimization Methods and Software Vol. 23, no. 5 (2008), p. 793-810
- Full Text: false
- Reviewed:
- Description: The classical Remez algorithm was developed for constructing the best polynomial approximations for continuous and discrete functions in an interval [a, b]. In this paper, the classical Remez algorithm is generalized to the problem of linear spline approximation with certain conditions on the spline parameters. Namely, the spline parameters have to be nonnegative and the values of the splines at one of the borders (or both borders) of the approximation intervals may be fixed. This type of constraint occurs in some practical applications, e.g. the problem of taxation tables restoration. The results of the numerical experiments with a Remez-like algorithm developed for this class of conditional optimization problems, are presented.
- Description: C1
Characterization theorem for best linear spline approximation with free knots
- Authors: Sukhorukova, Nadezda , Ugon, Julien
- Date: 2010
- Type: Text , Journal article
- Relation: Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms Vol. 17, no. 5 (2010), p. 687-708
- Full Text: false
- Reviewed:
- Description: A necessary condition for a best Chebyshev approximation by piecewise linear functions is derived using quasidifferential calculus. We first discover some properties of the knots joining the linear functions. Then we use these properties to obtain the optimality condition. This condition is stronger than existing results. We present an example of linear spline approximation where the existing optimality conditions are satisfied, but not the proposed one, which shows that it is not optimal. Copyright © 2010 Watam Press.
Workload coverage through nonsmooth optimization
- Authors: Sukhorukova, Nadezda , Ugon, Julien , Yearwood, John
- Date: 2009
- Type: Text , Journal article
- Relation: Optimization Methods and Software Vol. 24, no. 2 (2009), p. 285-298
- Full Text: false
- Reviewed:
- Description: In this paper, workload coverage is the problem of identifying a pattern of days worked and days off, along with the number of hours worked on each work day. This pattern must satisfy certain work-related constraints and fit best to a predefined workload. In our study, we formulate the problem of workload coverage as an optimization problem. We propose a number of models which take into consideration various staffing constraints. For each of these models, our study aims to find a compromise between an accurate workload coverage and the ability to solve the corresponding optimization problems in a reasonable time. Numerical experiments on each model are carried out and the results are presented. Interestingly, the nonlinear programming approaches are found to be competitive with linear programming ones. © 2009 Taylor & Francis.
Uniform approximation by the highest defect continuous polynomial splines : Necessary and sufficient optimality conditions and their generalisations
- Authors: Sukhorukova, Nadezda
- Date: 2010
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 147, no. 2 (2010), p. 378-394
- Full Text: false
- Reviewed:
- Description: In this paper necessary and sufficient optimality conditions for uniform approximation of continuous functions by polynomial splines with fixed knots are derived. The obtained results are generalisations of the existing results obtained for polynomial approximation and polynomial spline approximation. The main result is two-fold. First, the generalisation of the existing results to the case when the degree of the polynomials, which compose polynomial splines, can vary from one subinterval to another. Second, the construction of necessary and sufficient optimality conditions for polynomial spline approximation with fixed values of the splines at one or both borders of the corresponding approximation interval. © 2010 Springer Science+Business Media, LLC.
Minimisation of the cluster function : Numerical experiments using MPI techniques
- Authors: Sukhorukova, Nadezda
- Date: 2004
- Type: Text , Conference paper
- Relation: Paper presented at ICOTA6: 6th International Conference on Optimization - Techniques and Applications, Ballarat, Victoria : 9th December, 2004
- Full Text: false
- Reviewed:
- Description: Several clustering methods based on optimisation have been developed recently. One of them is based on minimisation of the cluster function. This function is nonsmooth, nonconvex and extremely multi extremal. Minimisation of such functions is a challenging task. This process can be also very time consuming, especially if the dimension of the corresponding optimisation problem and the size of the dataset are large. In this paper we propose an approach which allows one to run programs in parallel using several CPUs simultaneously. We discuss several possible ways for design parallel implementations for the program and present results of numerical experiments.
- Description: E1
- Description: 2003000934
Vallee poussin theorem and remez algorithm in the case of generalised degree polynomial spline approximation
- Authors: Sukhorukova, Nadezda
- Date: 2010
- Type: Text , Journal article
- Relation: Pacific Journal of Optimization Vol. 6, no. 1 (2010), p. 103-114
- Full Text: false
- Description: The classical Remez algorithm was developed for constructing the best polynomial approximations for continuous and discrete functions in an interval. In this paper the classical Remez algorithm is generalised to the problem of polynomial spline (piece-wise polynomial) approximation with the spline defect equal to the spline degree. Also, the values of the splines in the end points of the approximation interval may be fixed Polynomial splines combine simplicity of polynomials and flexibility, which allows one to significantly decrease the degree of the corresponding polynomials and oscillations of deviation functions. Therefore polynomial splines are a powerful tool for function and data approximation. The generalisation of the Remez algorithm developed in this research has been tested on several approximation problems. The results of the numerical experiments are presented.
A necessary optimality condition for free knots linear splines: Special cases
- Authors: Sukhorukova, Nadezda
- Date: 2010
- Type: Text , Journal article
- Relation: Pacific Journal of Optimization Vol. 6, no. 2, Suppl. 1 (2010), p. 305-317
- Full Text: false
- Description: In this paper, we study the problem of best Chebyshev approximation by linear splines. We construct linear splines as a max - min of linear functions. Then we apply nonsmooth optimisation techniques to analyse and solve the corresponding optimisation problems. This approach allows us to identify and introduce a new important property of linear spline knots (regular and irregular). Using this property, we derive a necessary optimality condition for the case of regular knots. This condition is stronger than those existing in the literature. We also present a numerical example which demonstrates the difference between the old and the new optimality conditions.
Chebyshev approximation by linear combinations of fixed knot polynomial splines with weighting functions
- Authors: Sukhorukova, Nadezda , Ugon, Julien
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 171, no. 2 (2016), p. 536-549
- Full Text: false
- Reviewed:
- Description: In this paper, we derive conditions for best uniform approximation by fixed knots polynomial splines with weighting functions. The theory of Chebyshev approximation for fixed knots polynomial functions is very elegant and complete. Necessary and sufficient optimality conditions have been developed leading to efficient algorithms for constructing optimal spline approximations. The optimality conditions are based on the notion of alternance (maximal deviation points with alternating deviation signs). In this paper, we extend these results to the case when the model function is a product of fixed knots polynomial splines (whose parameters are subject to optimization) and other functions (whose parameters are predefined). This problem is nonsmooth, and therefore, we make use of convex and nonsmooth analysis to solve it.
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.
Polynomial splines and data approximation
- Authors: Sukhorukova, Nadezda
- Date: 2003
- Type: Text , Conference paper
- Relation: Paper presented at the Symposium on Industrial Optimisation and the 9th Australian Optimisation Day, Perth : 30th September, 2002
- Full Text: false
- Reviewed:
- Description: The problem of data approximation is of great interest. There are a lot of approaches to solve this problem. One of them is a polynomial spline approximation. In this paper we propose a new algorithm for polynomial spline approximation based on nonsmooth optimization techniques. Numerical experiments using this algorithm have been carried out. The results are presented and discussed.
- Description: E1
- Description: 2003000352
K-complex detection using a hybrid-synergic machine learning method
- Authors: Vu, Huy Quan , Li, Gang , Sukhorukova, Nadezda , Beliakov, Gleb , Liu, Shaowu , Philippe, Carole , Amiel, Hélène , Ugon, Adrien
- Date: 2012
- Type: Text , Journal article
- Relation: IEEE Transactions on Systems, Man and Cybernetics Part C : Applications and Reviews Vol. 42, no. 6 (2012), p. 1478-1490
- Full Text: false
- Reviewed:
- Description: Sleep stage identification is the first step in modern sleep disorder diagnostics process. K-complex is an indicator for the sleep stage 2. However, due to the ambiguity of the translation of the medical standards into a computer-based procedure, reliability of automated K-complex detection from the EEG wave is still far from expectation. More specifically, there are some significant barriers to the research of automatic K-complex detection. First, there is no adequate description of K-complex that makes it difficult to develop automatic detection algorithm. Second, human experts only provided the label for whether a whole EEG segment contains K-complex or not, rather than individual labels for each subsegment. These barriers render most pattern recognition algorithms inapplicable in detecting K-complex. In this paper, we attempt to address these two challenges, by designing a new feature extraction method that can transform visual features of the EEG wave with any length into mathematical representation and proposing a hybrid-synergic machine learning method to build a K-complex classifier. The tenfold cross-validation results indicate that both the accuracy and the precision of this proposed model are at least as good as a human expert in K-complex detection. © 1998-2012 IEEE.
- Description: 2003010569