Abstract
We consider the maximization \(\gamma = \max\{x^{T}\!Ax : x\in \{-1, 1\}^n\}\) for a given symmetric \(A\in\mathcal{R}^{n\times n}\). It was shown recently, using properties of zonotopes and hyperplane arrangements, that in the special case that A has a small rank, so that A = VV T where \(V\in\mathcal{R}^{n\times m}\) with m < n, then there exists a polynomial time algorithm (polynomial in n for a given m) to solve the problem \(\max\{x^TV V^Tx : x\in \{-1, 1\}^n\}\). In this paper, we use this result, as well as a spectral decomposition of A to obtain a sequence of non-increasing upper bounds on γ with no constraints on the rank of A. We also give easily computable necessary and sufficient conditions for the absence of a gap between the solution and its upper bound. Finally, we incorporate the semidefinite relaxation upper bound into our scheme and give an illustrative example.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alizadeh, F.: Combinatorial optimization with semidefinite matrices. In: Proceedings of the second annual integer programming and combinatorial optimization conference, Carnegie–Mellon University (1992)
Allemand K., Fukuda K., Liebling T.M. and Steiner E. (2001). A polynomial case of unconstrained zero–one quadratic optimization. Math. Prog. Ser. A. 91: 49–52
Avis D. and Fukuda K. (1996). Reverse search for enumeration. Discrete Appl. Math. 65: 21–46
Balas E., Ceria S. and Cornuejols G. (1993). A lift-and-project cutting plane algorithm for mixed 0-1 programs. Math. Prog. 58: 295–324
Buck R.C. (1943). Partition of space. Am. Math. Monthly. 50: 541–544
Delorme C. and Poljak S. (1993). Laplacian eigenvalues and the maximum–cut problem. Math. Prog. 62(3): 557–574
Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer Verlag (1987)
Edelsbrunner H., O’Rourke J. and Seidel R. (1986). Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput. 15(2): 341–363
Ferrez, J.A., Fukuda, K., Liebling, T.M.: Cuts, zonotopes and arrangements. In: Festschrift, P. (ed.), The Sharpest Cut. SIAM Series on Optimization (2001)
Floudas, C.A., Pardalos, P.M., Adjiman, C.S., Esposito, W.R., Gumus, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers (1999)
Garey M.R. and Johnson D.S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman San Francisco, CA
Gungah, S.K.: Maximally Robust Controllers for a Class of Unstructured Uncertainty. PhD thesis, Imperial College, University of London (1999)
Gungah S.K., Halikias G.D. and Jaimoukha I.M. (2000). Maximally robust controllers for multivariable systems. SIAM J. Control and Optim. 38(6): 1805–1829
Gungah, S.K., Halikias, G.D., Jaimoukha, I.M.: Maximally robust controllers for multivariable systems. In: Proc. IEEE Conf. Dec. & Control, pp. 595–600 (2000)
Gungah, S.K., Malik, U., Jaimoukha, I.M., Halikias, G.D.: A new upper bound for the real structured singular value. In Proc. IEEE Conf. Dec. & Control, pp. 247–248, Florida (2001)
Jaimoukha I.M., Halikias G.D., Malik U. and Gungah S.K. (2006). On the gap between the complex structured singular value and its convex upper bound. SIAM J. Control Optim. 45(4): 1251–1278
Lovasz L. and Shrijver A. (1991). Cones of matrices and set–functions and 0–1 optimization. SIAM J Optim. 1(2): 166–190
Malik, U.: New Bounds for the Structured Singular Value. PhD thesis, Imperial College, University of London (2003)
Malik U., Jaimoukha I.M., Halikias G.D. and Gungah S.K. (2006). On the gap between the quadratic integer programming problem and its semidefinite relaxation. Math. Prog. Series A 107(3): 505–515
Onn S. and Schulman L.J. (2001). The vector partition problem for convex objective functions. Math. Oper. Res. 26(3): 583–590
Poljak S., Rendl F. and Wolkowicz H. (1995). A recipe for semidefinite relaxation for (0,1)-quadratic programming. J. Global Optim. 7(1): 51–73
Poljak S. and Wolkowicz H. (1982). Convex relaxation of (0,1)–quadratic programming. Math. Oper. Res. 20(3): 550–561
(2000). Handbook of Semidefinite Programming: Theory, Algorithms and Applications. Kluwer Academic Publishers, Boston, MA
Zaslavsky T. (1975). Facing up to arrangements: face–count formulas for partitions of space by hyperplanes. Mem. Am. Math. Soc. 154(5): 585–600
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Halikias, G.D., Jaimoukha, I.M., Malik, U. et al. New bounds on the unconstrained quadratic integer programming problem. J Glob Optim 39, 543–554 (2007). https://doi.org/10.1007/s10898-007-9155-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-007-9155-z