Abstract
This paper addresses the problem of finding tight affine lower bound functions for multivariate polynomials. Such underestimating functions are needed if global optimization problems involving polynomials are solved with a branch and bound method. These bound functions are constructed by using the expansion of the given polynomial into Bernstein polynomials. In contrast to our previous method which requires in the general case the solution of a linear programming problem, we propose here a method which requires only the solution of a system of linear equations together with a sequence of back substitutions and the computation of slopes. An error bound exhibiting quadratic convergence in the univariate case and some numerical examples are presented.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Berkelaar M., LP_SOLVE: Linear Programming Code. ftp://ftp.ics.ele.tue.nl/pub/lp_solve/
Cargo G. T. and Shisha O. (1966), “The Bernstein form of a polynomial,” J. Res. Nat. Bur. Standards Vol. 70B, 79–81.
Floudas C. A. (2000), “Deterministic Global Optimization: Theory, Methods, and Applications,” Series Nonconvex Optimization and its Applications Vol. 37, Kluwer Acad. Publ., Dordrecht, Boston, London.
Garloff J. (1986), “Convergent bounds for the range of multivariate polynomials,” Interval Mathematics 1985, K. Nickel, editor, Lecture Notes in Computer Science Vol. 212, Springer, Berlin, 37–56.
Garloff J., Jansson C. and Smith A. P. (2003), “Lower bound functions for polynomials,” J. Computational and Applied Mathematics, to appear.
Garloff J., Jansson C. and Smith A. P. (2003), “Inclusion isotonicity of convex-concave extensions for polynomials based on Bernstein expansion,” Computing, to appear.
Hansen, E. R. (1992), “Global Optimization Using Interval Analysis,” Marcel Dekker, Inc., New York.
Kearfott R. B. (1996), “Rigorous Global Search: Continuous Problems,” Series Non-convex Optimization and its Applications Vol. 13, Kluwer Acad. Publ., Dordrecht, Boston, London.
Liberti L. and Pantelides C. C. (2002), “Convex envelopes of monomials of odd degree,” J. Global Optimization Vol. 25, 157–168.
Prautzsch H., Boehm W. and Paluszny M. (2002), “Bézier and B-Spline Techniques,” Springer, Berlin, Heidelberg.
Ratschek H. and Rokne J. (1988), “New Computer Methods for Global Optimization,” Ellis Horwood Ltd., Chichester.
Tawarmalani M. and Sahinidis N. V. (2002), “Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications,” Series Nonconvex Optimization and its Applications Vol. 65, Kluwer Acad. Publ., Dordrecht, Boston, London.
Zettler M. and Garloff J. (1998), “Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion,” IEEE Trans. Automat. Contr. Vol. 43, 425–431.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Kluwer Academic Publishers
About this paper
Cite this paper
Garloff, J., Smith, A.P. (2004). An Improved Method for the Computation of Affine Lower Bound Functions for Polynomials. In: Floudas, C.A., Pardalos, P. (eds) Frontiers in Global Optimization. Nonconvex Optimization and Its Applications, vol 74. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0251-3_8
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0251-3_8
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7961-4
Online ISBN: 978-1-4613-0251-3
eBook Packages: Springer Book Archive