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Mathematical Programming

, Volume 170, Issue 1, pp 141–176 | Cite as

Strong formulations for quadratic optimization with M-matrices and indicator variables

  • Alper AtamtürkEmail author
  • Andrés Gómez
Full Length Paper Series B

Abstract

We study quadratic optimization with indicator variables and an M-matrix, i.e., a PSD matrix with non-positive off-diagonal entries, which arises directly in image segmentation and portfolio optimization with transaction costs, as well as a substructure of general quadratic optimization problems. We prove, under mild assumptions, that the minimization problem is solvable in polynomial time by showing its equivalence to a submodular minimization problem. To strengthen the formulation, we decompose the quadratic function into a sum of simple quadratic functions with at most two indicator variables each, and provide the convex-hull descriptions of these sets. We also describe strong conic quadratic valid inequalities. Preliminary computational experiments indicate that the proposed inequalities can substantially improve the strength of the continuous relaxations with respect to the standard perspective reformulation.

Keywords

Quadratic optimization Submodularity Perspective formulation Conic quadratic cuts Convex piecewise nonlinear inequalities 

Mathematics Subject Classification

90C11 90C20 90C57 

References

  1. 1.
    Ahuja, R.K., Hochbaum, D.S., Orlin, J.B.: A cut-based algorithm for the nonlinear dual of the minimum cost network flow problem. Algorithmica 39, 189–208 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aktürk, M.S., Atamtürk, A., Gürel, S.: A strong conic quadratic reformulation for machine-job assignment with controllable processing times. Oper. Res. Lett. 37, 187–191 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anstreicher, K.M.: On convex relaxations for quadratically constrained quadratic programming. Math. Program. 136, 233–251 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Atamtürk, A., Bhardwaj, A.: Network design with probabilistic capacities. Networks 71, 16–30 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Atamtürk, A., Gomez, A.: Submodularity in conic quadratic mixed 0–1 optimization. BCOL Research Report 16.02, IEOR, UC Berkeley. arXiv preprint arXiv:1705.05918 (2017)
  6. 6.
    Atamtürk, A., Jeon, H.: Lifted polymatroid for mean-risk optimization with indicator variables. BCOL Research Report 17.01, IEOR, UC Berkeley. \(\text{arXiv}\,\,\text{ preprint }\) arXiv:1705.05915 (2017)
  7. 7.
    Atamtürk, A., Narayanan, V.: Cuts for conic mixed integer programming. In: Fischetti, M., Williamson, D.P. (eds.) Proceedings of the 12th International IPCO Conference, pp. 16–29 (2007)Google Scholar
  8. 8.
    Balas, E.: Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. SIAM J. Algebr. Discrete Methods 6, 466–486 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Belotti, P., Góez, J.C., Pólik, 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: Numerical Analysis and Optimization, pp. 1–35. Springer (2015)Google Scholar
  10. 10.
    Bertsimas, D., King, A., Mazumder, R.: Best subset selection via a modern optimization lens. Ann. Stat. 44, 813–852 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bienstock, D.: Computational study of a family of mixed-integer quadratic programming problems. Math. Program. 74, 121–140 (1996)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Bienstock, D., Michalka, A.: Cutting-planes for optimization of convex functions over nonconvex sets. SIAM J. Optim. 24, 643–677 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Boland, N., Dey, S.S., Kalinowski, T., Molinaro, M., Rigterink, F.: Bounding the gap between the McCormick relaxation and the convex hull for bilinear functions. Math. Program. 162, 523–535 (2017a)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Boland, N., Gupte, A., Kalinowski, T., Rigterink, F., Waterer, H.: Extended formulations for convex hulls of graphs of bilinear functions. arXiv preprint arXiv:1702.04813 (2017b)
  15. 15.
    Bonami, P., Lodi, A., Tramontani, A., Wiese, S.: On mathematical programming with indicator constraints. Math. Program. 151, 191–223 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Trans. Pattern Anal. Mach. Intell. 23, 1222–1239 (2001)CrossRefGoogle Scholar
  17. 17.
    Ceria, S., Soares, J.: Convex programming for disjunctive convex optimization. Math. Program. 86, 595–614 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cornuejols, G., Tütüncü, R.: Optimization Methods in Finance, vol. 5. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  19. 19.
    Dong, H., Linderoth, J.: On valid inequalities for quadratic programming with continuous variables and binary indicators. In: Goemans, M., Correa, J. (eds.) Proceedings of IPCO 2013, pp. 169–180. Springer, Berlin (2013)Google Scholar
  20. 20.
    Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schönenheim, J. (eds.) Combinatorial Structures and Their Applications, pp. 69–87. Gordon and Breach, Philadelphia (1970)Google Scholar
  21. 21.
    Frangioni, A., Gentile, C.: Perspective cuts for a class of convex 0–1 mixed integer programs. Math. Program. 106, 225–236 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Frangioni, A., Gentile, C., Hungerford, J.: Decompositions of semidefinite matrices and the perspective reformulation of nonseparable quadratic programs. Report R-16-10, IASI, Rome (2016)Google Scholar
  23. 23.
    Gao, J., Li, D.: Cardinality constrained linear-quadratic optimal control. IEEE Trans. Autom. Control 56, 1936–1941 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Günlük, O., Linderoth, J.: Perspective reformulations of mixed integer nonlinear programs with indicator variables. Math. Program. 124, 183–205 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hijazi, H., Bonami, P., Cornuéjols, G., Ouorou, A.: Mixed-integer nonlinear programs featuring “on/off” constraints. Comput. Optim. Appl. 52, 537–558 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I: Fundamentals, vol. 305. Springer, Berlin (2013)zbMATHGoogle Scholar
  27. 27.
    Hochbaum, D.S.: Multi-label markov random fields as an efficient and effective tool for image segmentation, total variations and regularization. Numer. Math. Theory Methods Appl. 6, 169–198 (2013)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Ivănescu, P.L.: Some network flow problems solved with pseudo-boolean programming. Oper. Res. 13, 388–399 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Jeon, H., Linderoth, J., Miller, A.: Quadratic cone cutting surfaces for quadratic programs with on–off constraints. Discrete Optim. 24, 32–50 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Keilson, J., Styan, G.P.H.: Markov chains and M-matrices: inequalities and equalities. J. Math. Anal. Appl. 41, 439–459 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kılınç-Karzan, F., Yıldız, S.: Two-term disjunctions on the second-order cone. Math. Program. 154, 463–491 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kolmogorov, V., Zabin, R.: What energy functions can be minimized via graph cuts? IEEE Trans. Pattern Anal. Mach. Intell. 26, 147–159 (2004)CrossRefGoogle Scholar
  33. 33.
    Lobo, M.S., Fazel, M., Boyd, S.: Portfolio optimization with linear and fixed transaction costs. Ann. Oper. Res. 152, 341–365 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lovász, L.: Submodular functions and convexity. In: Bachem, A., Korte, B., Grötschel, M. (eds.) Mathematical Programming The State of the Art: Bonn 1982, pp. 235–257. Springer, Berlin (1983)CrossRefGoogle Scholar
  35. 35.
    Luedtke, J., Namazifar, M., Linderoth, J.: Some results on the strength of relaxations of multilinear functions. Math. Program. 136, 325–351 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Luedtke, J., D’Ambrosio, C., Linderoth, J., Schweiger, J.: Strong convex nonlinear relaxations of the pooling problem. arXiv preprint arXiv:1803.02955 (2018)
  37. 37.
    Luk, F.T., Pagano, M.: Quadratic programming with M-matrices. Linear Algebra Appl. 33, 15–40 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Mahajan, A., Leyffer, S., Linderoth, J., Luedtke, J., Munson, T.: Minotaur: A mixed-integer nonlinear optimization toolkit. ANL/MCS-P8010-0817, Argonne National Lab (2017)Google Scholar
  39. 39.
    Modaresi, S., Kılınç, M.R., Vielma, J.P.: Intersection cuts for nonlinear integer programming: convexification techniques for structured sets. Math. Program. 155, 575–611 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions I. Math. Program. 14, 265–294 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Orlin, J.B.: A faster strongly polynomial time algorithm for submodular function minimization. Math. Program. 118, 237–251 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Picard, J.C., Ratliff, H.D.: Minimum cuts and related problems. Networks 5, 357–370 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Plemmons, R.J.: M-matrix characterizations. I—nonsingular M-matrices. Linear Algebra Appl. 18, 175–188 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Poljak, S., Wolkowicz, H.: Convex relaxations of (0,1)-quadratic programming. Math. Oper. Res. 20, 550–561 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Stubbs, R.A., Mehrotra, S.: A branch-and-cut method for 0–1 mixed convex programming. Math. Program. 86, 515–532 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Vielma, J.P.: Small and strong formulations for unions of convex sets from the cayley embedding. To appear in Mathematical Programming, arXiv preprint arXiv:1704.03954 (2018)
  47. 47.
    Wei, D., Sestok, C.K., Oppenheim, A.V.: Sparse filter design under a quadratic constraint: low-complexity algorithms. IEEE Trans. Signal Process. 61, 857–870 (2013)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Wu, B., Sun, X., Li, D., Zheng, X.: Quadratic convex reformulations for semicontinuous quadratic programming. SIAM J. Optim. 27, 1531–1553 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Young, N.: The rate of convergence of a matrix power series. Linear Algebra Appl. 35, 261–278 (1981)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Department of Industrial Engineering and Operations ResearchUniversity of CaliforniaBerkeleyUSA
  2. 2.Department of Industrial Engineering , Swanson School of EngineeringUniversity of PittsburghPittsburghUSA

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