Abstract

We propose a feasible active set method for convex quadratic programming problems with nonnegativity constraints. This method is specifically designed to be embedded into a branch-and-bound algorithm for convex quadratic mixed- integer programming problems. The branch-and-bound algorithm generalizes the approach for unconstrained convex quadratic integer programming proposed by Buchheim, Caprara, and Lodi [Math. Program., 135 (2012), pp. 369--395] to the presence of linear constraints. The main feature of the latter approach consists of a sophisticated preprocessing phase, leading to a fast enumeration of the branch-and-bound nodes. Moreover, the feasible active set method takes advantage of this preprocessing phase and is well suited for reoptimization. Experimental results for randomly generated instances show that the new approach significantly outperforms the MIQP solver of \tt CPLEX 12.6 for instances with a small number of constraints.

Keywords

  1. integer programming
  2. quadratic programming
  3. global optimization

MSC codes

  1. 90C10
  2. 90C20
  3. 90C57

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
T. Achterberg, SCIP: solving constraint integer programs, Math. Program. Comput., 1 (2009), pp. 1--41.
2.
S. C. Agrawal, On mixed integer quadratic programs, Naval Res. Logist. Quart., 21 (1974), pp. 289--297.
3.
F. A. Al-Khayyal and C. Larsen, Global optimization of a quadratic function subject to a bounded mixed integer constraint set, Ann. Oper. Res., 25 (1990), pp. 169--180.
4.
D. P. Bertsekas, Projected Newton methods for optimization problems with simple constraints, SIAM J. Control Optim., 20 (1982), pp. 221--246, ŭldoi:10.1137/0320018.
5.
D. P. Bertsekas, Nonlinear Programming, Athena Scientific, Belmont, MA, 1999.
6.
D. Bienstock, Computational study of a family of mixed-integer quadratic programming problems, Math. Programming, 74 (1996), pp. 121--140.
7.
P. Bonami, L. Biegler, A. Conn, G. Cornuéjols, I. Grossmann, C. Laird, J. Lee, A. Lodi, F. Margot, N. Sawaya, and A. Wächter, An algorithmic framework for convex mixed integer nonlinear programs, Discrete Optim., 5 (2008), pp. 186--204.
8.
C. Buchheim, A. Caprara, and A. Lodi, An effective branch-and-bound algorithm for convex quadratic integer programming, Math. Program., 135 (2012), pp. 369--395.
9.
C. Buchheim and L. Trieu, Active set methods with reoptimization for convex quadratic integer programming, in Combinatorial Optimization, Lecture Notes in Comput. Sci. 8596, Springer, Cham, 2014, pp. 125--136.
10.
G. Cornuéjols and R. Tütüncü, Optimization Methods in Finance, Math. Finance Risk, Cambridge University Press, Cambridge, UK, 2006.
11.
R. S. Dembo, S. C. Eisenstat, and T. Steinhaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), pp. 400--408, ŭldoi:10.1137/0719025.
12.
E. Dolan and J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), pp. 201--213.
13.
F. Facchinei and S. Lucidi, Quadratically and superlinearly convergent algorithms for the solution of inequality constrained minimization problems, J. Optim. Theory Appl., 85 (1995), pp. 265--289.
14.
FICO Xpress Optimization Suite, 2015, www.fico.com.
15.
R. Fletcher and S. Leyffer, Numerical experience with lower bounds for MIQP branch-and-bound, SIAM J. Optim., 8 (1998), pp. 604--616, ŭldoi:10.1137/S1052623494268455.
16.
L. Grippo, F. Lampariello, and S. Lucidi, A truncated newton method with nonmonotone line search for unconstrained optimization, J. Optim. Theory Appl., 60 (1989), pp. 401--419.
17.
GUROBI Optimizer, 2015. www.gurobi.com.
18.
IBM ILOG CPLEX Optimizer, 2015. www.ibm.com/software/commerce/optimization/ cplex-optimizer.
19.
R. Lazimy, Mixed-integer quadratic programming, Math. Programming, 22 (1982), pp. 332--349.
20.
R. Lazimy, Improved algorithm for mixed-integer quadratic programs and a computational study, Math. Programming, 32 (1985), pp. 100--113.
21.
S. Leyffer, Deterministic Methods for Mixed Integer Nonlinear Programming, Ph.D. thesis, University of Dundee, Scotland, UK, 1993.
22.
MOSEK Optimization Software, 2015. mosek.com/products/mosek.
23.
J. Nocedal and S. Wright, Numerical Optimization, Springer-Verlag, New York, 1999.
24.
B. N. Pshenichny and Y. M. Danilin, Numerical Methods in Extremal Problems, Mir Publishers, Moscow, 1978.

Information & Authors

Information

Published In

cover image SIAM Journal on Optimization
SIAM Journal on Optimization
Pages: 1695 - 1714
ISSN (online): 1095-7189

History

Submitted: 23 July 2014
Accepted: 19 May 2016
Published online: 24 August 2016

Keywords

  1. integer programming
  2. quadratic programming
  3. global optimization

MSC codes

  1. 90C10
  2. 90C20
  3. 90C57

Authors

Affiliations

Funding Information

Deutsche Forschungsgemeinschaft http://dx.doi.org/10.13039/501100001659 : BU 2313/4

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media