Abstract
Maximization of a convex quadratic form on a convex polyhedral set is an NP-hard problem. We focus on computing an upper bound based on a factorization of the quadratic form matrix and employment of the maximum vector norm. Effectivity of this approach depends on the factorization used. We discuss several choices as well as iterative methods to improve performance of a particular factorization. We carried out numerical experiments to compare various alternatives and to compare our approach with other standard approaches, including McCormick envelopes.
Supported by the Czech Science Foundation Grant P403-18-04735S.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Allemand, K., Fukuda, K., Liebling, T.M., Steiner, E.: A polynomial case of unconstrained zero-one quadratic optimization. Math. Program. 91(1), 49–52 (2001)
Baharev, A., Achterberg, T., Rév, E.: Computation of an extractive distillation column with affine arithmetic. AIChE J. 55(7), 1695–1704 (2009)
Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming. Theory and Algorithms. 3rd edn. Wiley, Hoboken (2006)
Černý, M., Hladík, M.: The complexity of computation and approximation of the t-ratio over one-dimensional interval data. Comput. Stat. Data Anal. 80, 26–43 (2014)
Floudas, C.A.: Deterministic Global Optimization. Theory, Methods and Applications, Nonconvex Optimization and its Applications, vol. 37. Kluwer, Dordrecht (2000)
Floudas, C.A., Visweswaran, V.: Quadratic optimization. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp. 217–269. Springer, Boston (1995)
Gould, N.I.M., Toint, P.L.: A quadratic programming bibliography. RAL Internal Report 2000-1, Science and Technology Facilities Council, Scientific Computing Department, Numerical Analysis Group, 28 March, 2012. ftp://ftp.numerical.rl.ac.uk/pub/qpbook/qp.pdf
Hansen, E.R., Walster, G.W.: Global Optimization Using Interval Analysis, 2nd edn. Marcel Dekker, New York (2004)
Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Heidelberg (1990)
Konno, H.: Maximizing a convex quadratic function over a hypercube. J. Oper. Res. Soc. Jpn 23(2), 171–188 (1980)
Kreinovich, V., Lakeyev, A., Rohn, J., Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer, Dordrecht (1998)
McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I - Convex underestimating problems. Math. Program. 10(1), 147–175 (1976)
Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)
Pardalos, P., Rosen, J.: Methods for global concave minimization: a bibliographic survey. SIAM Rev. 28(3), 367–79 (1986)
Sherali, H.D., Adams, W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer, Boston (1999)
Tuy, H.: Convex Analysis and Global Optimization. Springer Optimization and Its Applications, vol. 110, 2nd edn. Springer, Cham (2016)
Vavasis, S.A.: Nonlinear Optimization: Complexity Issues. Oxford University Press, New York (1991)
Vavasis, S.A.: Polynomial time weak approximation algorithms for quadratic programming. In: Pardalos, P.M. (ed.) Complexity in Numerical Optimization, pp. 490–500. World Scientific Publishing, Singapore (1993)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Hladík, M., Hartman, D. (2020). Maximization of a Convex Quadratic Form on a Polytope: Factorization and the Chebyshev Norm Bounds. In: Le Thi, H., Le, H., Pham Dinh, T. (eds) Optimization of Complex Systems: Theory, Models, Algorithms and Applications. WCGO 2019. Advances in Intelligent Systems and Computing, vol 991. Springer, Cham. https://doi.org/10.1007/978-3-030-21803-4_12
Download citation
DOI: https://doi.org/10.1007/978-3-030-21803-4_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-21802-7
Online ISBN: 978-3-030-21803-4
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)