Abstract
We are concerned with tools to find bounds on the range of certain polynomial functions of n variables. Although our motivation and history of the tools are from crisp global optimization, bounding the range of such functions is also important in fuzzy logic implementations. We review and provide a new perspective on one such tool. We have been examining problems naturally posed in terms of barycentric coordinates, that is, over simplexes. There is a long history of using Bernstein expansions to bound ranges of polynomials over simplexes, particularly within the computer graphics community for 1-, 2-, and 3-dimensional problems, with some literature on higher-dimensional generalizations, and some work on use in global optimization. We revisit this work, identifying efficient implementation and practical application contexts, to bound ranges of polynomials over simplexes in dimensions, 2, 3, and higher.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Not to be confused with the standard simplex in \(\mathbb {R}^n\) of the literature, defined in terms of (2).
References
Bezerra, L.: Efficient computation of Bézier curves from their Bernstein-Fourier representation. Appl. Math. Comput. 220, 235–238 (2013)
Böhm, W., Farin, G., Kahmann, J.: A survey of curve and surface methods in CAGD. Comput. Aided Geom. Des. 1(1), 1–60 (1984)
Farouki, R.T.: The Bernstein polynomial basis: a centennial retrospective. Comput. Aided Geom. Des. 29(6), 379–419 (2012)
Hu, C., Kearfott, R.B., de Korvin, A. (eds.): Knowledge Processing with Interval and Soft Computing. Advanced Information and Knowledge Processing. Springer, New York (2008). https://doi.org/10.1007/978-1-84800-326-2
Karhbet, S., Kearfott, R.B.: Range bounds of functions over simplices, for branch and bound algorithms. Reliab. Comput. 25, 53–73 (2017). Special volume containing refereed papers from SCAN 2016, guest editors Vladik Kreinovich and Warwick Tucker
Kreinovich, V.: Relations between interval and soft computing, pp. 75–97. In: Hu, C. et al. [4] (2008)
Leroy, R.: Convergence under subdivision and complexity of polynomial minimization in the simplicial Bernstein basis. Reliab. Comput. 17(1), 11–21 (2012)
Lodwick, W.A., Jamison, K.D.: Special issue: interfaces between fuzzy set theory and interval analysis. Fuzzy Sets Syst. 135(1), 1–3 (2003). Interfaces between fuzzy set theory and interval analysis
Markowitz, H.: Portfolio selection. J. Finance 7(1), 77–91 (1952)
Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)
Moore, R., Lodwick, W.: Interval analysis and fuzzy set theory. Fuzzy Sets Syst. 135(1), 5–9 (2003). Interfaces between fuzzy set theory and interval analysis
Muñoz, C., Narkawicz, A.: Formalization of a representation of Bernstein polynomials and applications to global optimization. J. Autom. Reason. 51(2), 151–196 (2013)
Nataraj, P.S.V., Arounassalame, M.: An interval newton method based on the Bernstein form for bounding the zeros of polynomial systems. Reliab. Comput. 15(2), 109–119 (2011)
Peters, J.: Evaluation and approximate evaluation of the multivariate Bernstein-Bézier form on a regularly partitioned simplex. ACM Trans. Math. Softw. 20(4), 460–480 (1994)
Ray, S., Nataraj, P.S.V.: A new strategy for selecting subdivision point in the Bernstein approach to polynomial optimization. Reliab. Comput. 14(1), 117–137 (2010)
Ray, S., Nataraj, P.S.V.: A matrix method for efficient computation of Bernstein coefficients. Reliab. Comput. 17(1), 40–71 (2012)
So, A.M.-C.: Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems. Math. Program. 129(2), 357–382 (2011)
Tang, J.F., Wang, D.W., Fung, R.Y.K., Yung, K.-L.: Understanding of fuzzy optimization: theories and methods. J. Syst. Sci. Complex. 17(1), 117 (2004)
Titi, J., Garloff, J.: Fast determination of the tensorial and simplicial Bernstein forms of multivariate polynomials and rational functions. Reliab. Comput. 25, 24–37 (2017)
Titi, J., Garloff, J.: Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomials. Appl. Math. Comput. 315, 246–258 (2017)
Titi, J., Hamadneh, T., Garloff, J.: Convergence of the simplicial rational Bernstein form. In: Le Thi, H.A., Pham Dinh, T., Nguyen, N.T. (eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences. AISC, vol. 359, pp. 433–441. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-18161-5_37
Ullah, A., Li, J., Hussain, A., Shen, Y.: Genetic optimization of fuzzy membership functions for cloud resource provisioning. In: 2016 IEEE Symposium Series on Computational Intelligence (SSCI), pp. 1–8, December 2016
Waggenspack, W.N., Anderson, D.C.: Converting standard bivariate polynomials to Bernstein form over arbitrary triangular regions. Comput. Aided Des. 18(10), 529–532 (1986)
Walster, G.W., Kreinovich, V.: Computational complexity of optimization and crude range testing: a new approach motivated by fuzzy optimization. Fuzzy Sets Syst. 135(1), 179–208 (2003). Interfaces between fuzzy set theory and interval analysis
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Kearfott, R.B., Liu, D. (2018). A Brief Review of a Method for Bounds on Polynomial Ranges over Simplexes. In: Barreto, G., Coelho, R. (eds) Fuzzy Information Processing. NAFIPS 2018. Communications in Computer and Information Science, vol 831. Springer, Cham. https://doi.org/10.1007/978-3-319-95312-0_44
Download citation
DOI: https://doi.org/10.1007/978-3-319-95312-0_44
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-95311-3
Online ISBN: 978-3-319-95312-0
eBook Packages: Computer ScienceComputer Science (R0)