Outer-product-free sets for polynomial optimization and oracle-based cuts

Abstract

This paper introduces cutting planes that involve minimal structural assumptions, enabling the generation of strong polyhedral relaxations for a broad class of problems. We consider valid inequalities for the set \(S\cap P\), where S is a closed set, and P is a polyhedron. Given an oracle that provides the distance from a point to S, we construct a pure cutting plane algorithm which is shown to converge if the initial relaxation is a polyhedron. These cuts are generated from convex forbidden zones, or S-free sets, derived from the oracle. We also consider the special case of polynomial optimization. Accordingly we develop a theory of outer-product-free sets, where S is the set of real, symmetric matrices of the form \(xx^T\). All maximal outer-product-free sets of full dimension are shown to be convex cones and we identify several families of such sets. These families are used to generate strengthened intersection cuts that can separate any infeasible extreme point of a linear programming relaxation efficiently. Computational experiments demonstrate the promise of our approach.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

Notes

  1. 1.

    This BoxQP relaxation only adds the “diagonal” McCormick estimates \(X_{ii} \le x_i\).

References

  1. 1.

    Andersen, K., Jensen, A.N.: Intersection cuts for mixed integer conic quadratic sets. In: Goemans, M., Correa, J. (eds.) Integer Programming and Combinatorial Optimization, pp. 37–48. Springer, Berlin (2013)

    MATH  Google Scholar 

  2. 2.

    Andersen, K., Louveaux, Q., Weismantel, R.: An analysis of mixed integer linear sets based on lattice point free convex sets. Math. Oper. Res. 35(1), 233–256 (2010)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Andersen, K., Louveaux, Q., Weismantel, R., Wolsey, L.A.: Inequalities from two rows of a simplex tableau. In: Fischetti, M., Williamson, D.P. (eds.) Integer Programming and Combinatorial Optimization, pp. 1–15. Springer, Berlin (2007)

    Google Scholar 

  4. 4.

    Anstreicher, K.M.: Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming. J. Global Optim. 43(2–3), 471–484 (2009)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Atamtürk, A., Narayanan, V.: Conic mixed-integer rounding cuts. Math. Program. 122(1), 1–20 (2010)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Audet, C., Hansen, P., Jaumard, B., Savard, G.: A branch and cut algorithm for nonconvex quadratically constrained quadratic programming. Math. Program. 87(1), 131–152 (2000)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Averkov, G.: On finite generation and infinite convergence of generalized closures from the theory of cutting planes. (2011). arXiv preprint arXiv:1106.1526

  8. 8.

    Balas, E.: Intersection cuts–a new type of cutting planes for integer programming. Oper. Res. 19(1), 19–39 (1971)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. Discret. Appl. Math. 89(1–3), 3–44 (1998)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Balas, E., Saxena, A.: Optimizing over the split closure. Math. Program. 113(2), 219–240 (2008)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Bao, X., Sahinidis, N.V., Tawarmalani, M.: Multiterm polyhedral relaxations for nonconvex, quadratically constrained quadratic programs. Optim. Methods Softw. 24(4–5), 485–504 (2009)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Basu, A., Conforti, M., Cornuéjols, G., Zambelli, G.: Maximal lattice-free convex sets in linear subspaces. Math. Oper. Res. 35(3), 704–720 (2010)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Basu, A., Conforti, M., Cornuéjols, G., Zambelli, G.: Minimal inequalities for an infinite relaxation of integer programs. SIAM J. Discret. Math. 24(1), 158–168 (2010)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Basu, A., Cornuéjols, G., Zambelli, G.: Convex sets and minimal sublinear functions. J. Convex Anal. 18(2), 427–432 (2011)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Belotti, P., Góez, J.C., Pólik, I., Ralphs, T.K., Terlaky, T.: On families of quadratic surfaces having fixed intersections with two hyperplanes. Discret. Appl. Math. 161(16–17), 2778–2793 (2013)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-integer nonlinear optimization. Acta Numer. 22, 1–131 (2013)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. Numer. Math. 4(1), 238–252 (1962)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Bienstock, D., Michalka, A.: Cutting-planes for optimization of convex functions over nonconvex sets. SIAM J. Optim. 24(2), 643–677 (2014)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Bonami, P., Günlük, O., Linderoth, J.: Globally solving nonconvex quadratic programming problems with box constraints via integer programming methods. Math. Programm. Comput. 10(3), 333–382 (2018)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Borozan, V., Cornuéjols, G.: Minimal valid inequalities for integer constraints. Math. Oper. Res. 34(3), 538–546 (2009)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Burer, S.: Optimizing a polyhedral-semidefinite relaxation of completely positive programs. Math. Programm. Comput. 2(1), 1–19 (2010)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Burer, S., Vandenbussche, D.: Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound. Comput. Optim. Appl. 43, 181–195 (2009)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Chen, J., Burer, S.: Globally solving nonconvex quadratic programming problems via completely positive programming. Math. Programm. Comput. 4(1), 33–52 (2012)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discret. Math. 4(4), 305–337 (1973)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Conforti, M., Cornuéjols, G., Daniilidis, A., Lemaréchal, C., Malick, J.: Cut-generating functions and S-free sets. Math. Oper. Res. 40(2), 276–391 (2014)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Conforti, M., Cornuéjols, G., Zambelli, G.: Equivalence between intersection cuts and the corner polyhedron. Oper. Res. Lett. 38(3), 153–155 (2010)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Cornuéjols, G., Wolsey, L., Yıldız, S.: Sufficiency of cut-generating functions. Math. Program. 152(1–2), 643–651 (2015)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Dadush, D., Dey, S.S., Vielma, J.P.: The split closure of a strictly convex body. Oper. Res. Lett. 39(2), 121–126 (2011)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Dax, A.: Low-rank positive approximants of symmetric matrices. Adv. Linear Algebra Matrix Theory 4(3), 172–185 (2014)

    Google Scholar 

  30. 30.

    Del Pia, A., Weismantel, R.: On convergence in mixed integer programming. Math. Program. 135(1–2), 397–412 (2012)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Dey, S.S., Wolsey, L.A.: Lifting integer variables in minimal inequalities corresponding to lattice-free triangles. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) Integer Programming and Combinatorial Optimization, pp. 463–475. Springer, Berlin (2008)

    MATH  Google Scholar 

  32. 32.

    Dey, S.S., Wolsey, L.A.: Constrained infinite group relaxations of MIPs. SIAM J. Optim. 20(6), 2890–2912 (2010)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrika 1(3), 211–218 (1936)

    MATH  Google Scholar 

  34. 34.

    Fischetti, M., Ljubić, I., Monaci, M., Sinnl, M.: A new general-purpose algorithm for mixed-integer bilevel linear programs. Oper. Res. 65(6), 1615–1637 (2017)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Fischetti, M., Lodi, A.: Optimizing over the first Chvátal closure. Math. Program. 110(1), 3–20 (2007)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Fischetti, M., Salvagnin, D., Zanette, A.: A note on the selection of Benders’ cuts. Math. Program. 124(1–2), 175–182 (2010)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Floudas, C.A., Pardalos, P.M., Adjiman, C., Esposito, W.R., Gümüs, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of Test Problems in Local and Global Optimization, vol. 33. Springer Science & Business Media, Berlin (2013)

    MATH  Google Scholar 

  38. 38.

    Freund, R.M., Orlin, J.B.: On the complexity of four polyhedral set containment problems. Math. Program. 33(2), 139–145 (1985)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Ghaddar, B., Vera, J.C., Anjos, M.F.: A dynamic inequality generation scheme for polynomial programming. Math. Program. 156(1–2), 21–57 (2016)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Glover, F.: Polyhedral convexity cuts and negative edge extensions. Zeitschrift für Oper. Res. 18(5), 181–186 (1974)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bull. Am. Math. Soc. 64(5), 275–278 (1958)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Gomory, R.E.: An algorithm for integer solutions to linear programs. In: Graves, R.L., Wolfe, P. (eds.) Recent Advances in Mathematical Programming, pp. 269–302. McGraw-Hill, New York (1963)

    Google Scholar 

  43. 43.

    Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra. Math. Program. 3(1), 23–85 (1972)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2), 169–197 (1981)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Guennebaud, G., Jacob, B., et al.: Eigen v3. (2010). http://eigen.tuxfamily.org

  46. 46.

    Higham, N.J.: Computing a nearest symmetric positive semidefinite matrix. Linear Algebra Appl. 103, 103–118 (1988)

    MathSciNet  MATH  Google Scholar 

  47. 47.

    Hillestad, R.J., Jacobsen, S.E.: Reverse convex programming. Appl. Math. Optim. 6(1), 63–78 (1980)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Hiriart-Urruty, J.B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer Science & Business Media, Berlin (2012)

    MATH  Google Scholar 

  49. 49.

    Kelley Jr., J.E.: The cutting-plane method for solving convex programs. J. Soc. Ind. Appl. Math. 8(4), 703–712 (1960)

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Kılınç-Karzan, F.: On minimal valid inequalities for mixed integer conic programs. Math. Oper. Res. 41(2), 477–510 (2015)

    MathSciNet  MATH  Google Scholar 

  51. 51.

    Kocuk, B., Dey, S.S., Sun, X.A.: Matrix minor reformulation and SOCP-based spatial branch-and-cut method for the AC optimal power flow problem. Math. Programm. Comput. 10(4), 557–596 (2018)

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Konno, H., Yamamoto, R.: Choosing the best set of variables in regression analysis using integer programming. J. Global Optim. 44(2), 273–282 (2009)

    MathSciNet  MATH  Google Scholar 

  53. 53.

    Krishnan, K., Mitchell, J.E.: A unifying framework for several cutting plane methods for semidefinite programming. Optim. Methods Softw. 21(1), 57–74 (2006)

    MathSciNet  MATH  Google Scholar 

  54. 54.

    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)

    MathSciNet  MATH  Google Scholar 

  55. 55.

    Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Emerging applications of algebraic geometry, pp 157–270. Springer, Berlin (2009)

  56. 56.

    Li, Y., Richard, J.P.P.: Cook, Kannan and Schrijver’s example revisited. Discrete Optim. 5(4), 724–734 (2008)

    MathSciNet  MATH  Google Scholar 

  57. 57.

    Locatelli, M., Schoen, F.: On convex envelopes for bivariate functions over polytopes. Math. Program. 144(1–2), 65–91 (2014)

    MathSciNet  MATH  Google Scholar 

  58. 58.

    Locatelli, M., Thoai, N.V.: Finite exact branch-and-bound algorithms for concave minimization over polytopes. J. Global Optim. 18(2), 107–128 (2000)

    MathSciNet  MATH  Google Scholar 

  59. 59.

    Lovász, L.: Geometry of numbers and integer programming. Mathematical Programming: Recent Developments and Applications pp. 177–210 (1989)

  60. 60.

    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1(2), 166–190 (1991)

    MathSciNet  MATH  Google Scholar 

  61. 61.

    Luedtke, J., Namazifar, M., Linderoth, J.: Some results on the strength of relaxations of multilinear functions. Math. Program. 136(2), 325–351 (2012)

    MathSciNet  MATH  Google Scholar 

  62. 62.

    Marchand, H., Wolsey, L.A.: Aggregation and mixed integer rounding to solve MIPs. Oper. Res. 49(3), 363–371 (2001)

    MathSciNet  MATH  Google Scholar 

  63. 63.

    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I–Convex underestimating problems. Math. Program. 10(1), 147–175 (1976)

    MATH  Google Scholar 

  64. 64.

    Meeraus, A.: GLOBALLib. http://www.gamsworld.org/global/globallib.htm

  65. 65.

    Mirsky, L.: Symmetric gauge functions and unitarily invariant norms. Q. J. Math. 11(1), 50–59 (1960)

    MathSciNet  MATH  Google Scholar 

  66. 66.

    Misener, R., Floudas, C.A.: Global optimization of mixed-integer quadratically-constrained quadratic programs (MIQCQP) through piecewise-linear and edge-concave relaxations. Math. Program. 136(1), 155–182 (2012)

    MathSciNet  MATH  Google Scholar 

  67. 67.

    Misener, R., Smadbeck, J.B., Floudas, C.A.: Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2. Optim. Methods and Softw. 30(1), 215–249 (2015)

    MathSciNet  MATH  Google Scholar 

  68. 68.

    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)

    MathSciNet  MATH  Google Scholar 

  69. 69.

    Modaresi, S., Kılınç, M.R., Vielma, J.P.: Intersection cuts for nonlinear integer programming: convexification techniques for structured sets. Math. Program. 155(1–2), 575–611 (2016)

    MathSciNet  MATH  Google Scholar 

  70. 70.

    MOSEK ApS: The MOSEK Fusion API for C++ 8.1.0.63 (2018). https://docs.mosek.com/8.1/cxxfusion/index.html

  71. 71.

    Padberg, M., Rinaldi, G.: A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems. SIAM Rev. 33(1), 60–100 (1991)

    MathSciNet  MATH  Google Scholar 

  72. 72.

    Porembski, M.: Cone adaptation strategies for a finite and exact cutting plane algorithm for concave minimization. J. Global Optim. 24(1), 89–107 (2002)

    MathSciNet  MATH  Google Scholar 

  73. 73.

    Qualizza, A., Belotti, P., Margot, F.: Linear programming relaxations of quadratically constrained quadratic programs. In: Mixed Integer Nonlinear Programming pp. 407–426 (2012)

  74. 74.

    Rikun, A.D.: A convex envelope formula for multilinear functions. J. Global Optim. 10(4), 425–437 (1997)

    MathSciNet  MATH  Google Scholar 

  75. 75.

    Rockafellar, R.T.: Convex Analysis, vol. 28. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  76. 76.

    Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations. Math. Program. 124(1–2), 383–411 (2010)

    MathSciNet  MATH  Google Scholar 

  77. 77.

    Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations. Math. Program. 130(2), 359–413 (2011)

    MathSciNet  MATH  Google Scholar 

  78. 78.

    Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, vol. 151, 2nd edn. Cambridge University Press, Cambridge (2014)

    MATH  Google Scholar 

  79. 79.

    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1986)

    MATH  Google Scholar 

  80. 80.

    Sen, S., Sherali, H.D.: Nondifferentiable reverse convex programs and facetial convexity cuts via a disjunctive characterization. Math. Program. 37(2), 169–183 (1987)

    MathSciNet  MATH  Google Scholar 

  81. 81.

    Serrano, F.: Intersection cuts for factorable MINLP. In: Lodi, A., Nagarajan, V. (eds.) Integer Programming and Combinatorial Optimization, pp. 385–398. Springer International Publishing, Berlin (2019)

    Google Scholar 

  82. 82.

    Sherali, H.D., Fraticelli, B.M.P.: Enhancing RLT relaxations via a new class of semidefinite cuts. J. Global Optim. 22(1–4), 233–261 (2002)

    MathSciNet  MATH  Google Scholar 

  83. 83.

    Shor, N.Z.: Quadratic optimization problems. Sov. J. Comput. Syst. Sci. 25, 1–11 (1987)

    MathSciNet  MATH  Google Scholar 

  84. 84.

    Tardella, F.: Existence and sum decomposition of vertex polyhedral convex envelopes. Optim. Lett. 2(3), 363–375 (2008)

    MathSciNet  MATH  Google Scholar 

  85. 85.

    Tawarmalani, M., Richard, J.P.P., Xiong, C.: Explicit convex and concave envelopes through polyhedral subdivisions. Math. Program. 138(1–2), 531–577 (2013)

    MathSciNet  MATH  Google Scholar 

  86. 86.

    Tawarmalani, M., Sahinidis, N.V.: Convex extensions and envelopes of lower semi-continuous functions. Math. Program. 93(2), 247–263 (2002)

    MathSciNet  MATH  Google Scholar 

  87. 87.

    Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications, vol. 65. Springer Science & Business Media, Berlin (2002)

    MATH  Google Scholar 

  88. 88.

    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)

    MathSciNet  MATH  Google Scholar 

  89. 89.

    Towle, E., Luedtke, J.: Intersection disjunctions for reverse convex sets. (2019). arXiv preprint arXiv:1901.02112

  90. 90.

    Tuy, H.: Concave programming under linear constraints. Sov. Math. 5, 1437–1440 (1964)

    MATH  Google Scholar 

  91. 91.

    Vandenbussche, D., Nemhauser, G.: A branch-and-cut algorithm for nonconvex quadratic programs with box constraints. Math. Program. 102(3), 559–575 (2005)

    MathSciNet  MATH  Google Scholar 

  92. 92.

    Wolsey, L.A., Nemhauser, G.L.: Integer and Combinatorial Optimization. Wiley, New York (2014)

    MATH  Google Scholar 

  93. 93.

    Xia, W., Vera, J.C., Zuluaga, L.F.: Globally solving nonconvex quadratic programs via linear integer programming techniques. INFORMS J. Comput. (2019)

Download references

Acknowledgements

The authors thank Eli Towle for pointing out an error in the presentation of the intersection cut strengthening procedure, Felipe Serrano for useful comments and suggestions that led to Lemma 6, and to the anonymous reviewers whose thorough feedback greatly improved the article. The authors would also like to thank the Institute for Data Valorization (IVADO) for their support through the IVADO Postdoctoral Fellowship program.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Gonzalo Muñoz.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Conicyt BCH 72130388; G. Muñoz. ONR N00014-16-1-2889; D. Bienstock.

Appendices

Appendix

A radius of the conic hull of a ball

Suppose we have a ball of radius r and with centre that is distance \(m>r\) from the origin. We wish to determine the radius of the conic hull of the ball at a specific point along its axis. Consider a 2-dimensional cross-section of the conic hull of the ball containing the axis; this is shown in Fig. 4 in rectangular (xy) coordinates. A line passing through the origin and tangent to the boundary of the ball in the nonnegative orthant may be written in the form \(y=ax\) for some \(a>0\); let \((\bar{r}, \bar{m})\) be the point of intersection between line and ball. At \((\bar{r}, \bar{m})\) we have

$$\begin{aligned} (a\bar{r}-m)^2+\bar{r}^2=r^2 \iff (1+a^2)\bar{r}^2-2am\bar{r} + m^2-r^2 = 0. \end{aligned}$$
(21)

Now Eq. (21) should only have one unique solution with respect to \(\bar{r}\) since the line is tangent to the ball; thus the discriminant must be zero,

$$\begin{aligned} 4a^2m^2-4(1+a^2)(m^2-r^2)=0 \implies a=\frac{\sqrt{m^2-r^2}}{r}. \end{aligned}$$
(22)

Solving Eq. (21) for \(\bar{r}\) with Eq. (22),

$$\begin{aligned} \bar{r}= & {} \frac{2am}{2(1+a^2)}, \\= & {} \frac{r}{m}\sqrt{m^2-r^2},\\ \bar{m}= & {} a\bar{r}, \\= & {} \frac{m^2-r^2}{m}. \end{aligned}$$

Hence at distance d from the origin along the axis of the cone, the radius of the cone is \(\frac{\bar{r}}{\bar{m}}d\), or

$$\begin{aligned} \frac{r}{\sqrt{m^2-r^2}}d. \end{aligned}$$
(23)
Fig. 4
figure4

In grey, a ball with radius r and distance \(m>r\) from the origin. In red, the boundary of its conic hull. In black, an intersection point between the boundary of the ball and its conic hull (color figure online)

B Additional BoxQP experiments

See Table 5.

Table 5 Comparison of intersection cuts and wRLT+SDP on larger BoxQP instances

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bienstock, D., Chen, C. & Muñoz, G. Outer-product-free sets for polynomial optimization and oracle-based cuts. Math. Program. 183, 105–148 (2020). https://doi.org/10.1007/s10107-020-01484-3

Download citation

Keywords

  • Polynomial optimization
  • Intersection cuts
  • Cutting planes
  • S-free sets

Mathematics Subject Classification

  • 90C30
  • 90C57