Abstract
We present semidefinite relaxations for unconstrained non-convex quadratic mixed-integer optimization problems. These relaxations yield tight bounds and are computationally easy to solve for medium-sized instances, even if some of the variables are integer and unbounded. In this case, the problem contains an infinite number of linear constraints; these constraints are separated dynamically. We use this approach as a bounding routine in an SDP-based branch-and-bound framework. In case of a convex objective function, the new SDP bound improves the bound given by the continuous relaxation of the problem. Numerical experiments show that our algorithm performs well on various types of non-convex instances.
Similar content being viewed by others
References
Anstreicher K.: Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming. J. Glob. Optim. 43(2–3), 471–484 (2009)
Belotti, P.: Couenne: a user’s manual. Technical report, Lehigh University (2009)
Bonami P., Biegler L.T., Conn A.R., Cornuéjols G., Grossmann I.E., Laird C.D., Lee J., Lodi A., Margot F., Sawaya N., Wächter A.: An algorithmic framework for convex mixed integer nonlinear programs. Discret. Optim. 5, 186–2004 (2008)
Borchers B.: CSDP, a C library for semidefinite programming. Optim. Methods Softw. 11/12(1–4), 613–623 (1999)
Branke J., Scheckenbach B., Stein M., Deb K., Schmeck H.: Portfolio optimization with an envelope-based multi-objective evolutionary algorithm. Eur. J. Oper. Res. 199(3), 684–693 (2009)
Buchheim, C., Caprara, A., Lodi, A.: An effective branch-and-bound algorithm for convex quadratic integer programming. Math. Program. (2012, to appear)
Burer S.: Optimizing a polyhedral-semidefinite relaxation of completely positive programs. Math. Program. Comput. 2(1), 1–19 (2010) ISSN 1867-2949
Goemans M., Williamson D.: Improved approximation algorithms for maximum cut and satisfiability problems. J. ACM 42, 1115–1145 (1995)
ILOG, Inc. ILOG CPLEX 12.1 (2009) http://www.ilog.com/products/cplex
McCormick G.P.: Computability of global solutions to factorable nonconvex programs. I. Convex underestimating problems. Math. Program. 10(2), 147–175 (1976) ISSN 0025-5610
Phuong N.T.H., Tuy H., Al-Khayyal F.: Optimization of a quadratic function with a circulant matrix. Comput. Optim. Appl. 35(2), 135–159 (2006)
Qualizza, A., Belotti, P., Margot, F.: Linear programming relaxations of quadratically constrained quadratic programs. In: IMA Volume Series, pp. 407–426. Springer (2012)
Rendl F., Rinaldi G., Wiegele A.: Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Math. Program. 121(2), 307–335 (2010)
Ryoo H.S., Sahinidis N.V.: A branch-and-reduce approach to global optimization. J. Glob. Optim. 8(2), 107–138 (1996)
Sahinidis, N.V., Tawarmalani, M.: BARON 9.0.4: global optimization of mixed-integer nonlinear programs. User’s manual (2010)
Saxena A., Bonami P., Lee J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations. Math. Program. 124(1–2), 383–411 (2010)
Sherali H.D., Adams W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems, vol. 31 of Nonconvex Optimization and Its Applications. Kluwer, Dordrecht (1999)
Vandenberghe L., Boyd S.: Semidefinite programming. SIAM Rev. 38, 49–95 (1996)
Vandenbussche D., Nemhauser G.L.: A branch-and-cut algorithm for nonconvex quadratic programs with box constraints. Math. Program. 102(3), 559–575 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Buchheim, C., Wiegele, A. Semidefinite relaxations for non-convex quadratic mixed-integer programming. Math. Program. 141, 435–452 (2013). https://doi.org/10.1007/s10107-012-0534-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-012-0534-y