Abstract
Determining global integer extrema of an real-valued box-constrained multivariate quadratic functions is a very difficult task. In this paper, we present an analytic method, which is based on a combinatorial optimization approach in order to calculate global integer extrema of a real-valued box-constrained multivariate quadratic function, whereby this problem will be proven to be as NP-hard via solving it by a Travelling Salesman instance. Instead, we solve it using eigenvalue theory, which allows us to calculate the eigenvalues of an arbitrary symmetric matrix using Newton’s method, which converges quadratically and in addition yields a Jordan normal form with \(1 \times 1\)-blocks, from which a special representation of the multivariate quadratic function based on affine linear functions can be derived. Finally, global integer minimizers can be calculated dynamically and efficiently most often in a small amount of time using the Fourier–Motzkin- and a Branch and Bound like Dijkstra-algorithm. As an application, we consider a box-constrained bivariate and multivariate quadratic function with ten arguments.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Applegate DL, Bixby RM, Chvátal V, Cook WJ (2006) The traveling salesman problem. ISBN 0-691-12993-2
Artin E (1960/61) Analytic Geometry und Algebra I, II. Lessons at he university of Hamburg, 1960/61. Part I worked out by H. Behncke und W. Hansen, Part II worked out by H. Kiendl und W. Hansen
Arora S (1998) Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J ACM 45(5):753–782
Beardwood J, Halton JH, Hammersley JM (1959) The shortest path through many points. Proc Camb Philos Soc 55:299–327
Bellman R (1960) Combinatorial processes and dynamic programming. In: Bellman R, Hall M Jr (eds) Combinatorial analysis, Proceedings of symposia in applied mathematics 10. American Mathematical Society, Providence, pp 217–249
Bellman R (1962) Dynamic programming treatment of the travelling salesman problem. J Assoc Comput Mach 9:61–63
Berman P, Karpinski M (2006) 8/7-Approximation algorithm for (1,2)-TSP. In: Proceedings of 17th ACM-SIAM symposium on discrete algorithms (SODA ’06), pp 641–648
Bosch S (2001) Algebra, 4th edn. Springer, Berlin
Chi L (2004) Extrema of a real polynomial. J Glob Optim 30(4):405–433
Christofides N (1976) Worst-case analysis of a new heuristic for the travelling salesman problem. Technical report 388, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh
Dantzig GB, Fulkerson R, Johnson SM (1954) Solution of a large-scale traveling salesman problem. Oper Res 2(4):393–410
Dijkstra W (1959) A note on two problems in connexion with graphs. Numer Math 1:269–271
Gutin G, Yeo A, Zverovich A (2002) Traveling salesman should not be greedy: domination analysis of greedy-type heuristics for the TSP. Discrete Appl Math 117(13):81–86
Hassin R, Rubinstein S (2000) Better approximations for max TSP. Inf Process Lett 75(4):181–186
Held M, Karp RM (1962) A dynamic programming approach to sequencing problems. J Soc Ind Appl Math 10(1):196–210
Householder AS (1970) The numerical treatment of a single nonlinear Equation. MacGraw-Hill, New York
Kaplan H, Lewenstein L, Shafrir N, Sviridenko M (2004) Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs. In: Proceedings of 44th IEEE symposium on foundations of computer science, pp 56–65
Karpinski M, Lampis M, Schmied R (2015) New Inapproximability bounds for TSP. J Comput Syst Sci 81(8):1665–1677
Lasserre JB (2001) Global optimization with polynomials and the problem of moments. SIAM J Optim 11(7):796–817
Lawler EL, Lenstra JK, Rinnooy Kan AHG, Shmoys DB (1985) The traveling salesman problem: a guided tour of combinatorial optimization. Wiley, London
Marden M (1966) Geometry of polynomials. American Mathematical Society, Providence
Padberg M, Rinaldi G (1991) A Branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems. SIAM Rev 33(1):60–100. doi:10.1137/1033004
Papadimitriou Christos H (1977) The Euclidean traveling salesman problem is NP-complete. Theor Comput Sci 4(3):237–244
Qi L, Teo KL (2003) Multivariate polynomial minimization and its application in signal processing. J Glob Optim 26:419–433
Rosenkrantz DJ, Stearns RE, Lewis PM II (1977) An analysis of several heuristics for the traveling salesman problem. SIAM J Comput 6(5):563–581
Shor NZ (1998) Non-differentiable optimization and polynomial problems. Kluwer, Dordrecht
Shustin E (1996) Critical points of real polynomials, subdivisions of Newton polyhedra and topology of real algebraic hyper-surfaces. Am Math Soc Transl 173:203–223
Steinerberger S (2015) New bounds for the traveling salesman constant. Adv Appl Probab 47:27–36
Thng I, Cantoni A, Leung YH (1996) Analytical solutions to the optimization of a quadratic cost function subject to linear and quadratic equality constraints. Appl Math Optim 34:161–382
Walshaw C (2000) A multilevel approach to the travelling salesman problem. CMS Press, London
Walshaw C (2001) A multilevel Lin–Kernighan–Helsgaun algorithm for the travelling salesman problem. CMS Press, London
Wörz S (2015) Optimising logistic problems in a traditional corn harvest chain. Dissertation, Chair for Agricultural Engineering of University of Technology Munich
Wöz S (2016) Solving traveling salesman problems using Dijkstra’s algorithm. Complexity (submitted)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wörz, S. On global integer extrema of real-valued box-constrained multivariate quadratic functions. J Comb Optim 34, 964–986 (2017). https://doi.org/10.1007/s10878-017-0123-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-017-0123-3
Keywords
- Combinatorial optimization
- Global optimization
- Global integer extrema
- Real-valued-box-constrained multivariate quadratic functions