Abstract
We present a unifying framework for generating extended formulations for the polyhedral outer approximations used in algorithms for mixed-integer convex programming (MICP). Extended formulations lead to fewer iterations of outer approximation algorithms and generally faster solution times. First, we observe that all MICP instances from the MINLPLIB2 benchmark library are conic representable with standard symmetric and nonsymmetric cones. Conic reformulations are shown to be effective extended formulations themselves because they encode separability structure. For mixed-integer conic-representable problems, we provide the first outer approximation algorithm with finite-time convergence guarantees, opening a path for the use of conic solvers for continuous relaxations. We then connect the popular modeling framework of disciplined convex programming (DCP) to the existence of extended formulations independent of conic representability. We present evidence that our approach can yield significant gains in practice, with the solution of a number of open instances from the MINLPLIB2 benchmark library.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
- 2.
Solutions reported to Stefan Vigerske, October 5, 2015.
References
MINLPLIB2 library. http://www.gamsworld.org/minlp/minlplib2/html/
Abhishek, K., Leyffer, S., Linderoth, J.: FilMINT: An outer approximation-based solver for convex mixed-integer nonlinear programs. INFORMS J. Comput. 22, 555–567 (2010)
Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1, 1–41 (2009)
Ahmadi, A., Olshevsky, A., Parrilo, P., Tsitsiklis, J.: NP-hardness of deciding convexity of quartic polynomials and related problems. Math. Program. 137, 453–476 (2013)
Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-Integer nonlinear optimization. Acta Numerica 22, 1–131 (2013)
Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2001)
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. Discrete Optim. 5, 186–204 (2008)
Bonami, P., Kilinç, M., Linderoth, J.: Algorithms and software for convex mixed integer nonlinear programs. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming. The IMA Volumes in Mathematics and its Applications, vol. 154, pp. 1–39. Springer, New York (2012)
Byrd, R.H., Nocedal, J., Waltz, R.: KNITRO: An integrated package for nonlinear optimization. In: di Pillo, G., Roma, M. (eds.) Large-Scale Nonlinear Optimization. Nonconvex Optimization and its Applications, vol. 83, pp. 35–59. Springer, Berlin (2006)
Diamond, S., Chu, E., Boyd, S.: Disciplined convex programming. http://dcp.stanford.edu/
Drewes, S., Ulbrich, S.: Subgradient based outer approximation for mixed integer second order cone programming. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming. The IMA Volumes in Mathematics and its Applications, vol. 154, pp. 41–59. Springer, New York (2012)
Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer approximation. Math. Program. 66, 327–349 (1994)
Goldberg, N., Leyffer, S.: An active-set method for second-order conic-constrained quadratic programming. SIAM J. Optim. 25, 1455–1477 (2015)
Grant, M., Boyd, S., Ye, Y.: Disciplined convex programming. In: Liberti, L., Maculan, N. (eds.) Global Optimization. Nonconvex Optimization and its Applica-tions, vol. 84, pp. 155–210. Springer, US (2006)
Günlük, O., Linderoth, J.: Perspective reformulation and applications. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming. The IMA Volumes in Mathematics and its Applications, vol. 154, pp. 61–89. Springer, New York (2012)
Gupta, O.K., Ravindran, A.: Branch and bound experiments in convex nonlinear integer programming. Manag. Sci. 31, 1533–1546 (1985)
Harjunkoski, I., Westerlund, T., Pörn, R., Skrifvars, H.: Different transformations for solving non-convex trim-loss problems by MINLP. Eur. J. Oper. Res. 105, 594–603 (1998)
Hien, L.: Differential properties of euclidean projection onto power cone. Math. Methods Oper. Res. 83(3), 265–284 (2015)
Hijazi, H., Bonami, P., Ouorou, A.: An outer-inner approximation for separable mixed-integer nonlinear programs. INFORMS J. Comput. 26, 31–44 (2014)
Kılınç, M.R.: Disjunctive cutting planes and algorithms for convex mixed integer nonlinear programming. Ph.D. thesis, University of Wisconsin-Madison (2011)
Leyffer, S.: Deterministic methods for mixed integer nonlinear programming. Ph.D. thesis, University of Dundee, December 1993
Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284, 193–228 (1998). International Linear Algebra Society (ILAS) Symposium on Fast Algorithms for Control, Signals and Image Processing
Lubin, M., Dunning, I.: Computing in operations research using Julia. INFORMS J. Comput. 27, 238–248 (2015)
Lubin, M., Yamangil, E., Bent, R., Vielma, J.P.: Extended formulations in mixed-integer convex programming, ArXiv e-prints (2015)
Mittelmann, H.: MINLP benchmark. http://plato.asu.edu/ftp/minlp_old.html
O’Donoghue, B., Chu, E., Parikh, N., Boyd, S.: Operator splitting for conic optimization via homogeneous self-dual embedding, ArXiv e-prints (2013)
Serrano, S.A.: Algorithms for unsymmetric cone optimization and an implementation for problems with the exponential cone. Ph.D. thesis, Stanford University, Stanford, CA, March 2015
Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)
Udell, M., Mohan, K., Zeng, D., Hong, J., Diamond, S., Boyd, S.: Convex optimization in Julia. In: Proceedings of HPTCDL 2014, Piscataway, NJ, USA, pp. 18–28. IEEE Press (2014)
Vielma, J.P., Dunning, I., Huchette, J., Lubin, M.: Extended formulations in mixed integer conic quadratic programming, ArXiv e-prints (2015)
Acknowledgements
We thank the anonymous referees for their comments. They greatly improved the clarity of the manuscript. We also thank one of the anonymous referees for pointing out the SOC-representability of the sssd family of instances originally derived in [15]. M. Lubin was supported by the DOE Computational Science Graduate Fellowship, which is provided under grant number DE-FG02-97ER25308. The work at LANL was funded by the Center for Nonlinear Studies (CNLS) and was carried out under the auspices of the NNSA of the U.S. DOE at LANL under Contract No. DE-AC52-06NA25396. J.P. Vielma was funded by NSF grant CMMI-1351619.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Lubin, M., Yamangil, E., Bent, R., Vielma, J.P. (2016). Extended Formulations in Mixed-Integer Convex Programming. In: Louveaux, Q., Skutella, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 2016. Lecture Notes in Computer Science(), vol 9682. Springer, Cham. https://doi.org/10.1007/978-3-319-33461-5_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-33461-5_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-33460-8
Online ISBN: 978-3-319-33461-5
eBook Packages: Computer ScienceComputer Science (R0)