Abstract
This paper deals with outer approximation based approaches to solve mixed integer second order cone programs. Thereby the outer approximation is based on subgradients of the second order cone constraints. Using strong duality of the subproblems that are solved during the algorithm, we are able to determine subgradients satisfying the KKT optimality conditions. This enables us to extend convergence results valid for continuously differentiable mixed integer nonlinear problems to subdifferentiable constraint functions. Furthermore, we present a version of the branch-and-bound based outer approximation that converges when relaxing the convergence assumption that every SOCP satisfies the Slater constraint qualification. We give numerical results for some application problems showing the performance of our approach.
AMS(MOS) subject classifications. 90C11.
Research partially supported by the German Research Foundation (DFG) within the SFB 805 and by the state of Hesse within the LOEWE-Center AdRIA.
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References
R. Fletcher and S. Leyffer, Solving Mixed Integer Nonlinear Programs by Outer Approximation, in Mathematical Programming, 1994, 66: 327–349.
R.A. Stubbs and S. Mehrotra, A branch-and-cut method for 0–1 mixed convex programming in Mathematical Programming, 1999, 86: 515–532.
I. Quesada and I.E. Grosmann, An LP/NLP based Branch and Bound Algorithm for Convex MINLP Optimization Problems, in Computers and Chemical Engineering, 1992, 16(10, 11): 937–947.
A.M. Geoffrion, Generalized Benders Decomposition, in Journal of Optimization Theory and Applications, 1972, 10(4): 237–260.
M.T. C¸ezik and G. Iyengar, Cuts for Mixed 0–1 Conic Programming, in Mathematical Programming, Ser. A, 2005, 104: 179–200.
M.A. Duran and I.E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs, in Mathematical Programming, 1986, 36: 307–339.
F. Alizadeh and D. Goldfarb, Second-Order Cone Programming, RUTCOR, Rutgers Center for Operations Research, Rutgers University, New Jersey, 2001.
P. Bonami, L.T. Biegler, A.R. Conn, G. Cornuejols, I.E. Grossmann, C.D. Laird, J. Lee, A. Lodi, F. Margot, N. Sawaya, and A. W¨achter, An Algorithmic Framework for Convex Mixed Integer Nonlinear Programs, IBM Research Division, New York, 2005.
R.A. Stubbs and S. Mehrotra, Generating Convex Polynomial Inequalities for Mixed 0–1 Programs, Journal of global optimization, 2002, 24: 311–332.
J.P. Vielma, S. Ahmed, and G.L. Nemhauser, A Lifted Linear Programming Branch-and-Bound Algorithm for Mixed Integer Conic Quadratic Programs, INFORMS Journal on Computing, 2008, 20(3): 438–450.
A. Atamt¨urk and V. Narayanan, Lifting for Conic Mixed-Integer Programming, BCOL Research report 07.04, 2007.
A. Atamt¨urk and V. Narayanan, Cuts for Conic Mixed-Integer Programming, Mathematical Programming, Ser. A, DOI 10.1007/s10107-008-0239-4, 2007.
A. Ben-Tal and A. Nemirovski, On Polyhedral Approximations of the Second- Order Cone, in Mathematics of Operations Research, 2001, 26(2): 193–205.
E. Balas, S. Ceria, and G. Cornu´ejols, A lift-and-project cutting plane algorithm for mixed 0–1 programs, in Mathematical Programming, 1993, 58: 295–324.
M. Fampa and N. Maculan, A new relaxation in conic form for the Euclidean Steiner Tree Problem in Rn, in RAIRO Operations Research, 2001, 35: 383–394.
J. Soukup and W.F. Chow, Set of test problems for the minimum length connection networks, in ACM SIGMAP Bulletin, 1973, 15: 48–51.
D. Bertsimas and R. Shioda, Algorithm for cardinality-constrained quadratic optimization, in Computational Optimization and Applications, 2007, 91: 239–269.
Y. Nesterov and A. Nemirovskii, Interior-Point Polynomial Algorithms in Convex Programming, SIAM Studies in Applied Mathematics, 2001.
R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970.
C. Geiger and C. Kanzow, Theorie und Numerik restringierter Optimierungsaufgaben, Springer Verlag Berlin Heidelberg New York, 2002.
J.E. Beasley, OR Library: Collection of test data for Euclidean Steiner Tree Problems, http://people.brunel.ac.uk/_mastjjb/jeb/orlib/esteininfo.html.
P. Belotti, P. Bonami, J.J. Forrest, L. Ladanyi, C. Laird, J. Lee, F. Margot, and A. W¨achter, BonMin, http://www.coin-or.org/Bonmin/.
R. Fletcher and S. Leyffer, User Manual of filterSQP, http://www.mcs.anl.gov/_leyffer/papers/SQP manual.pdf.
C. Laird and A. W¨achter, IPOPT, https://projects.coin-or.org/Ipopt.
K. Abhishek, S. Leyffer, and J.T. Linderoth, FilMINT: An Outer Approximation-Based Solver for Nonlinear Mixed Integer Programs, Argonne National Laboratory, Mathematics and Computer Science Division,2008.
S. Drewes, Mixed Integer Second Order Cone Programming, PhD Thesis, June, 2009.
P. Bonami, M. Kilinc, and J. Linderoth, Algorithms and Software for Convex Mixed Integer Nonlinear Programs 2009.
J.P. Vielma, Portfolio Optimization Instances http://www2.isye.gatech.edu/_jvielma/portfolio/.
S. Drewes,MISOCP Test Instances https://www3.mathematik.tu-darmstadt.de/index.php?id=491.
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Drewes, S., Ulbrich, S. (2012). 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. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1927-3_2
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DOI: https://doi.org/10.1007/978-1-4614-1927-3_2
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