Mathematical Programming

, Volume 162, Issue 1–2, pp 393–429 | Cite as

How to convexify the intersection of a second order cone and a nonconvex quadratic

Full Length Paper Series A

Abstract

A recent series of papers has examined the extension of disjunctive-programming techniques to mixed-integer second-order-cone programming. For example, it has been shown—by several authors using different techniques—that the convex hull of the intersection of an ellipsoid, \(\mathcal {E}\), and a split disjunction, \((l - x_j)(x_j - u) \le 0\) with \(l < u\), equals the intersection of \(\mathcal {E}\) with an additional second-order-cone representable (SOCr) set. In this paper, we study more general intersections of the form \(\mathcal {K}\cap \mathcal {Q}\) and \(\mathcal {K}\cap \mathcal {Q}\cap H\), where \(\mathcal {K}\) is a SOCr cone, \(\mathcal {Q}\) is a nonconvex cone defined by a single homogeneous quadratic, and H is an affine hyperplane. Under several easy-to-verify conditions, we derive simple, computable convex relaxations \(\mathcal {K}\cap \mathcal {S}\) and \(\mathcal {K}\cap \mathcal {S}\cap H\), where \(\mathcal {S}\) is a SOCr cone. Under further conditions, we prove that these two sets capture precisely the corresponding conic/convex hulls. Our approach unifies and extends previous results, and we illustrate its applicability and generality with many examples.

Keywords

Convex hull Disjunctive programming Mixed-integer linear programming Mixed-integer nonlinear programming Mixed-integer quadratic programming Nonconvex quadratic programming Second-order-cone programming Trust-region subproblem 

Mathematics Subject Classification

90C25 90C10 90C11 90C20 90C26 

Supplementary material

10107_2016_1045_MOESM1_ESM.pdf (904 kb)
Supplementary material 1 (pdf 903 KB)

References

  1. 1.
    Adjiman, C., Dallwig, S., Floudas, C., Neumaier, A.: A global optimization method, \(\alpha \)-BB, for general twice-differentiable constrained NLPs—I. Theoretical advances. Comput. Chem. Eng. 22(9), 1137–1158 (1998)CrossRefGoogle Scholar
  2. 2.
    Andersen, K., Jensen, A.N.: Intersection cuts for mixed integerconic quadratic sets. In: Proceedings of IPCO 2013, volume7801 of Lecture Notes in Computer Science, pp. 37–48.Valparaiso, Chile (March 2013)Google Scholar
  3. 3.
    Androulakis, I.P., Maranas, C.D., Floudas, C.A.: \(\alpha {{\rm BB}}\): a global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7(4), 337–363 (1995). State of the art in global optimization: computational methods and applications (Princeton, NJ, 1995)Google Scholar
  4. 4.
    Anstreicher, K.M., Burer, S.: Computable representations for convex hulls of low-dimensional quadratic forms. Math. Program. 124(1–2), 33–43 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Atamtürk, A., Narayanan, V.: Conic mixed-integer rounding cuts. Math. Program. 122(1), 1–20 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Balas, E.: Intersection cuts—a new type of cutting planes for integer programming. Oper. Res. 19, 19–39 (1971)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Balas, E.: Disjunctive programming. Ann. Discret. Math. 5, 3–51 (1979)CrossRefMATHGoogle Scholar
  8. 8.
    Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58, 295–324 (1993)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bao, X., Sahinidis, N.V., Tawarmalani, M.: Semidefinite relaxations for quadratically constrained quadratic programming: a review and comparisons. Math. Program. 129(1), 129–157 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Barvinok, A.: A Course in Convexity, vol. 54. American Mathematical Society, Providence (2002)MATHGoogle Scholar
  11. 11.
    Belotti, P.: Disjunctive cuts for nonconvex MINLP. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, volume 154 of The IMA Volumes in Mathematics and its Applications, pp. 117–144. Springer, New York, NY (2012)Google Scholar
  12. 12.
    Belotti, P., Góez, J., Pólik, I., Ralphs, T., Terlaky, T.: On families of quadratic surfaces having fixed intersections with two hyperplanes. Discret. Appl. Math. 161(16), 2778–2793 (2013)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Belotti, P., Goez, J.C., Polik, I., Ralphs, T.K., Terlaky, T.: A conic representation of the convex hull of disjunctive sets and conic cuts for integer second order cone optimization. In: Al-Baali, M., Grandinetti, L., Purnama, A. (eds.) Numerical Analysis and Optimization, volume 134 of Springer Proceedings in Mathematics and Statistics, pp. 1–35. Springer (2014)Google Scholar
  14. 14.
    Bienstock, D., Michalka, A.: Cutting-planes for optimization of convex functions over nonconvex sets. SIAM J. Optim. 24(2), 643–677 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Bonami, P.: Lift-and-project cuts for mixed integer convex programs. In: Gunluk, O., Woeginger, G.J. (eds.) Proceedings of the 15th IPCO Conference, volume 6655 of Lecture Notes in Computer Science, pp. 52–64. Springer, New York, NY (2011)Google Scholar
  16. 16.
    Burer, S., Anstreicher, K.M.: Second-order-cone constraints for extended trust-region subproblems. SIAM J. Optim. 23(1), 432–451 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Burer, S., Saxena, A.: The MILP road to MIQCP. In: Mixed Integer Nonlinear Programming, pp. 373–405. Springer (2012)Google Scholar
  18. 18.
    Cadoux, F.: Computing deep facet-defining disjunctive cuts for mixed-integer programming. Math. Program. 122(2), 197–223 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Çezik, M., Iyengar, G.: Cuts for mixed 0–1 conic programming. Math. Program. 104(1), 179–202 (2005)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Ceria, S., Soares, J.: Convex programming for disjunctive convex optimization. Math. Program. 86(3), 595–614 (1999)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-RegionMethods. MPS/SIAM Series on Optimization. SIAM, Philadelphia, PA (2000)Google Scholar
  22. 22.
    Cornuéjols, G., Lemaréchal, C.: A convex-analysis perspective on disjunctive cuts. Math. Program. 106(3), 567–586 (2006)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Dadush, D., Dey, S.S., Vielma, J.P.: The split closure of a strictly convex body. Oper. Res. Lett. 39, 121–126 (2011)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Drewes, S.: Mixed Integer Second Order Cone Programming. Ph.D. thesis, Technische Universität Darmstadt (2009)Google Scholar
  25. 25.
    Drewes, S., Pokutta, S.: Cutting-planes for weakly-coupled 0/1 second order cone programs. Electron. Notes in Discrete Math. 36, 735–742 (2010)CrossRefMATHGoogle Scholar
  26. 26.
    Gould, N.I.M., Lucidi, S., Roma, M., Toint, P.L.: Solving the trust-region subproblem using the Lanczos method. SIAM J. Optim. 9(2), 504–525 (1999)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Günlük, O., Linderoth, J.: Perspective reformulations of mixed integer nonlinear programs with indicator variables. Math. Program. 124(1–2), 183–205 (2010)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2013)MATHGoogle Scholar
  29. 29.
    Hu, J., Mitchell, J.E., Pang, J.-S., Bennett, K.P., Kunapuli, G.: On the global solution of linear programs with linear complementarity constraints. SIAM J. Optim. 19(1), 445–471 (2008)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Jeyakumar, V., Li, G.Y.: Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimization. Math. Program. 147(1), 171–206 (2013)MathSciNetMATHGoogle Scholar
  31. 31.
    Júdice, J.J., Sherali, H., Ribeiro, I.M., Faustino, A.M.: A complementarity-based partitioning and disjunctive cut algorithm for mathematical programming problems with equilibrium constraints. J. Glob. Optim. 136, 89–114 (2006)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Kato, T.: Perturbation Theory for Linear Operators, second edn. Springer, Berlin-New York (1976). Grundlehren der Mathematischen Wissenschaften, Band 132Google Scholar
  33. 33.
    Kılınç, M., Linderoth, J., Luedtke, J.: Effective separation of disjunctive cuts for convex mixed integer nonlinear programs. Technical report. http://www.optimization-online.org/DB_FILE/2010/11/2808.pdf (2010)
  34. 34.
    Kılınç-Karzan, F.: On minimal inequalities for mixed integer conic programs. Math. Oper. Res. 41(2), 477–510 (2016)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Kılınç-Karzan, F., Yıldız, S.: Two-term disjunctions on the second-order cone. In: Lee, J., Vygen, J. (eds.) IPCO, volume 8494 of Lecture Notes in Computer Science, pp. 345–356. Springer (2014)Google Scholar
  36. 36.
    Kılınç-Karzan, F., Yıldız, S.: Two-term disjunctions on the second-order cone. Math. Program. 154(1), 463–491 (2015)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Kim, S., Kojima, M.: Second order cone programming relaxation of nonconvex quadratic optimization problems. Optim. Methods Softw. 15(3–4), 201–224 (2001)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Mahajan, A., Munson, T.: Exploiting second-order cone structure for global optimization. Technical report. ANL/MCS-P1801-1010, Argonne National Laboratory, http://www.optimization-online.org/DB_HTML/2010/10/2780.html (October 2010)
  39. 39.
    Modaresi, S., Kılınç, M.R., Vielma, J.P.: Split cuts and extended formulations for mixed integer conic quadratic programming. Oper. Res. Lett. 43(1), 10–15 (2015)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Modaresi, S., Kılınç, M.R., Vielma, J.P.: Intersection cuts for nonlinear integer programming: convexification techniques for structured sets. Math. Program. 155(1), 575–611 (2016)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Modaresi, S., Vielma, J.: Convex hull of two quadratic or a conic quadratic and a quadratic inequality. Technical report. http://www.optimization-online.org/DB_HTML/2014/11/4641.html (November 2014)
  42. 42.
    Moré, J.J., Sorensen, D.C.: Computing a trust region step. SIAM J. Sci. Stat. Comput. 4(3), 553–572 (1983)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Nguyen, T.T., Tawarmalani, M., Richard, J.-P.P.: Convexification techniques for linear complementarity constraints. In: Günlük, O., Woeginger, G.J. (eds.) IPCO, volume 6655 of Lecture Notes in Computer Science, pp. 336–348. Springer (2011)Google Scholar
  44. 44.
    Pataki, G.: On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Math. Oper. Res. 23(2), 339–358 (1998)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Pólik, I., Terlaky, T.: A survey of the S-lemma. SIAM Rev. 49(3), 371–418 (2007). (electronic)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Rellich, F.: Perturbation theory of eigenvalue problems. Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz. Gordon and Breach Science Publishers, New York-London-Paris (1969)Google Scholar
  47. 47.
    Rendl, F., Wolkowicz, H.: A semidefinite framework for trust region subproblems with applications to large scale minimization. Math. Program. 77(2), 273–299 (1997)MathSciNetMATHGoogle Scholar
  48. 48.
    Saxena, A., Bonami, P., Lee, J.: Disjunctive cuts for non-convex mixed integer quadratically constrained programs. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO, volume 5035 of Lecture Notes in Computer Science, pp. 17–33. Springer (2008)Google Scholar
  49. 49.
    Sherali, H., Shetty, C.: Optimization with disjunctive constraints. Lectures on Econ. Math. Systems, 181 (1980)Google Scholar
  50. 50.
    Stubbs, R.A., Mehrotra, S.: A branch-and-cut method for 0–1 mixed convex programming. Math. Program. 86(3), 515–532 (1999)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Tawarmalani, M., Richard, J., Chung, K.: Strong valid inequalities for orthogonal disjunctions and bilinear covering sets. Math. Program. 124(1–2), 481–512 (2010)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Tawarmalani, M., Richard, J.-P.P., Xiong, C.: Explicit convex and concave envelopes through polyhedral subdivisions. Math. Program. 138(1–2), 531–577 (2013)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Vielma, J.P., Ahmed, S., Nemhauser, G.L.: A lifted linear programming branch-and-bound algorithm for mixed-integer conic quadratic programs. INFORMS J. Comput. 20(3), 438–450 (2008)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Yıldıran, U.: Convex hull of two quadratic constraints is an LMI set. IMA J. Math. Control Inf. 26, 417–450 (2009)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Yıldız, S., Cornuéjols, G.: Disjunctive cuts for cross-sections of the second-order cone. Oper. Res. Lett. 43(4), 432–437 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  1. 1.Department of Management SciencesUniversity of IowaIowa CityUSA
  2. 2.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA

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