Extending a CIP Framework to Solve MIQCPs
This paper discusses how to build a solver for mixed integer quadratically constrained programs (MIQCPs) by extending a framework for constraint integer programming (CIP). The advantage of this approach is that we can utilize the full power of advanced MILP and CP technologies, in particular for the linear relaxation and the discrete components of the problem. We use an outer approximation generated by linearization of convex constraints and linear underestimation of nonconvex constraints to relax the problem. Further, we give an overview of the reformulation, separation, and propagation techniques that are used to handle the quadratic constraints efficiently. We implemented these methods in the branch-cut-and-price framework SCIP. Computational experiments indicating the potential of the approach and evaluating the impact of the algorithmic components are provided.
Key wordsMixed integer quadratically constrained programming constraint integer programming branch-and-cut convex relaxation domain propagation primal heuristic nonconvex
Unable to display preview. Download preview PDF.
- 2.T. Achterberg, Constraint Integer Programming, PhD thesis, Technische Universit ¨at Berlin, 2007.Google Scholar
- 3., SCIP: Solving Constraint Integer Programs, Math. Program. Comput.,1 (2009), pp. 1–41.Google Scholar
- 4.T. Achterberg, T. Berthold, T. Koch, and K. Wolter, Constraint integer programming: A new approach to integrate CP and MIP, in Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, 5th International Conference, CPAIOR 2008, L. Perron and M. Trick, eds., Vol. 5015 of LNCS, Springer, 2008, pp. 6–20.Google Scholar
- 6.P. Belotti, J. Lee, L. Liberti, F. Margot, and A. W¨achter, Branching and bounds tightening techniques for non-convex MINLP, Optimization Methods and Software, 24 (2009), pp. 597–634.Google Scholar
- 8.M. B´enichou, J. M. Gauthier, P. Girodet, G. Hentges, G. Ribi`ere, and O. Vincent, Experiments in mixed-integer linear programming, Math. Program., 1 (1971), pp. 76–94.Google Scholar
- 9.T. Berthold, Primal heuristics for mixed integer programs. Diploma thesis, Technische Universit¨at Berlin, 2006.Google Scholar
- 10., RENS – relaxation enforced neighborhood search, ZIB-Report 07–28, Zuse Institute Berlin, 2007.Google Scholar
- 11.T. Berthold, S. Heinz, and M.E. Pfetsch, Nonlinear pseudo-boolean optimization: relaxation or propagation?, in Theory and Applications of Satisfiability Testing – SAT 2009, O. Kullmann, ed., no. 5584 in LNCS, Springer, 2009, pp. 441–446.Google Scholar
- 12.R.E. Bixby, M. Fenelon, Z. Gu, E. Rothberg, and R. Wunderling, MIP: theory and practice – closing the gap, in System Modelling and Optimization: Methods, Theory and Applications, M. Powell and S. Scholtes, eds., Kluwer, 2000, pp. 19–50.Google Scholar
- 13.P. Bonami, L.T. Biegler, A.R. Conn, G. Cornu´ejols, I.E. Grossmann, C.D. Laird, J. Lee, A. Lodi, F. Margot, N.W. Sawaya, and A. W¨achter, An algorithmic framework for convex mixed integer nonlinear programs, Discrete Optim., 5 (2008), pp. 186–204.Google Scholar
- 15.CMU-IBM MINLP Project. http://egon.cheme.cmu.edu/ibm/page.htm.
- 18.R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, Springer, 1990.Google Scholar
- 22.H. Mittelmann, MIQP test instances. http://plato.asu.edu/ftp/miqp.html.
- 23.MOSEK Corporation, The MOSEK optimization tools manual, 6.0 ed., 2009.Google Scholar
- 24.A. Saxena, P. Bonami, and J. Lee, Convex relaxations of non-convex mixed integer quadratically constrained programs: Projected formulations, Tech. Rep. RC24695, IBM Research, 2008. to appear in Math. Program.Google Scholar
- 26.M. Tawarmalani and N. Sahinidis, Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications, Kluwer Academic Publishers, 2002.Google Scholar
- 28.A. W¨achter and L.T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), pp. 25–57.Google Scholar