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.
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Notes
This BoxQP relaxation only adds the “diagonal” McCormick estimates \(X_{ii} \le x_i\).
References
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)
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)
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)
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)
Atamtürk, A., Narayanan, V.: Conic mixed-integer rounding cuts. Math. Program. 122(1), 1–20 (2010)
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)
Averkov, G.: On finite generation and infinite convergence of generalized closures from the theory of cutting planes. (2011). arXiv preprint arXiv:1106.1526
Balas, E.: Intersection cuts–a new type of cutting planes for integer programming. Oper. Res. 19(1), 19–39 (1971)
Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. Discret. Appl. Math. 89(1–3), 3–44 (1998)
Balas, E., Saxena, A.: Optimizing over the split closure. Math. Program. 113(2), 219–240 (2008)
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)
Basu, A., Conforti, M., Cornuéjols, G., Zambelli, G.: Maximal lattice-free convex sets in linear subspaces. Math. Oper. Res. 35(3), 704–720 (2010)
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)
Basu, A., Cornuéjols, G., Zambelli, G.: Convex sets and minimal sublinear functions. J. Convex Anal. 18(2), 427–432 (2011)
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)
Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-integer nonlinear optimization. Acta Numer. 22, 1–131 (2013)
Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. Numer. Math. 4(1), 238–252 (1962)
Bienstock, D., Michalka, A.: Cutting-planes for optimization of convex functions over nonconvex sets. SIAM J. Optim. 24(2), 643–677 (2014)
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)
Borozan, V., Cornuéjols, G.: Minimal valid inequalities for integer constraints. Math. Oper. Res. 34(3), 538–546 (2009)
Burer, S.: Optimizing a polyhedral-semidefinite relaxation of completely positive programs. Math. Programm. Comput. 2(1), 1–19 (2010)
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)
Chen, J., Burer, S.: Globally solving nonconvex quadratic programming problems via completely positive programming. Math. Programm. Comput. 4(1), 33–52 (2012)
Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discret. Math. 4(4), 305–337 (1973)
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)
Conforti, M., Cornuéjols, G., Zambelli, G.: Equivalence between intersection cuts and the corner polyhedron. Oper. Res. Lett. 38(3), 153–155 (2010)
Cornuéjols, G., Wolsey, L., Yıldız, S.: Sufficiency of cut-generating functions. Math. Program. 152(1–2), 643–651 (2015)
Dadush, D., Dey, S.S., Vielma, J.P.: The split closure of a strictly convex body. Oper. Res. Lett. 39(2), 121–126 (2011)
Dax, A.: Low-rank positive approximants of symmetric matrices. Adv. Linear Algebra Matrix Theory 4(3), 172–185 (2014)
Del Pia, A., Weismantel, R.: On convergence in mixed integer programming. Math. Program. 135(1–2), 397–412 (2012)
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)
Dey, S.S., Wolsey, L.A.: Constrained infinite group relaxations of MIPs. SIAM J. Optim. 20(6), 2890–2912 (2010)
Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrika 1(3), 211–218 (1936)
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)
Fischetti, M., Lodi, A.: Optimizing over the first Chvátal closure. Math. Program. 110(1), 3–20 (2007)
Fischetti, M., Salvagnin, D., Zanette, A.: A note on the selection of Benders’ cuts. Math. Program. 124(1–2), 175–182 (2010)
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)
Freund, R.M., Orlin, J.B.: On the complexity of four polyhedral set containment problems. Math. Program. 33(2), 139–145 (1985)
Ghaddar, B., Vera, J.C., Anjos, M.F.: A dynamic inequality generation scheme for polynomial programming. Math. Program. 156(1–2), 21–57 (2016)
Glover, F.: Polyhedral convexity cuts and negative edge extensions. Zeitschrift für Oper. Res. 18(5), 181–186 (1974)
Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bull. Am. Math. Soc. 64(5), 275–278 (1958)
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)
Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra. Math. Program. 3(1), 23–85 (1972)
Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2), 169–197 (1981)
Guennebaud, G., Jacob, B., et al.: Eigen v3. (2010). http://eigen.tuxfamily.org
Higham, N.J.: Computing a nearest symmetric positive semidefinite matrix. Linear Algebra Appl. 103, 103–118 (1988)
Hillestad, R.J., Jacobsen, S.E.: Reverse convex programming. Appl. Math. Optim. 6(1), 63–78 (1980)
Hiriart-Urruty, J.B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer Science & Business Media, Berlin (2012)
Kelley Jr., J.E.: The cutting-plane method for solving convex programs. J. Soc. Ind. Appl. Math. 8(4), 703–712 (1960)
Kılınç-Karzan, F.: On minimal valid inequalities for mixed integer conic programs. Math. Oper. Res. 41(2), 477–510 (2015)
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)
Konno, H., Yamamoto, R.: Choosing the best set of variables in regression analysis using integer programming. J. Global Optim. 44(2), 273–282 (2009)
Krishnan, K., Mitchell, J.E.: A unifying framework for several cutting plane methods for semidefinite programming. Optim. Methods Softw. 21(1), 57–74 (2006)
Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)
Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Emerging applications of algebraic geometry, pp 157–270. Springer, Berlin (2009)
Li, Y., Richard, J.P.P.: Cook, Kannan and Schrijver’s example revisited. Discrete Optim. 5(4), 724–734 (2008)
Locatelli, M., Schoen, F.: On convex envelopes for bivariate functions over polytopes. Math. Program. 144(1–2), 65–91 (2014)
Locatelli, M., Thoai, N.V.: Finite exact branch-and-bound algorithms for concave minimization over polytopes. J. Global Optim. 18(2), 107–128 (2000)
Lovász, L.: Geometry of numbers and integer programming. Mathematical Programming: Recent Developments and Applications pp. 177–210 (1989)
Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1(2), 166–190 (1991)
Luedtke, J., Namazifar, M., Linderoth, J.: Some results on the strength of relaxations of multilinear functions. Math. Program. 136(2), 325–351 (2012)
Marchand, H., Wolsey, L.A.: Aggregation and mixed integer rounding to solve MIPs. Oper. Res. 49(3), 363–371 (2001)
McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I–Convex underestimating problems. Math. Program. 10(1), 147–175 (1976)
Meeraus, A.: GLOBALLib. http://www.gamsworld.org/global/globallib.htm
Mirsky, L.: Symmetric gauge functions and unitarily invariant norms. Q. J. Math. 11(1), 50–59 (1960)
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)
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)
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)
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)
MOSEK ApS: The MOSEK Fusion API for C++ 8.1.0.63 (2018). https://docs.mosek.com/8.1/cxxfusion/index.html
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)
Porembski, M.: Cone adaptation strategies for a finite and exact cutting plane algorithm for concave minimization. J. Global Optim. 24(1), 89–107 (2002)
Qualizza, A., Belotti, P., Margot, F.: Linear programming relaxations of quadratically constrained quadratic programs. In: Mixed Integer Nonlinear Programming pp. 407–426 (2012)
Rikun, A.D.: A convex envelope formula for multilinear functions. J. Global Optim. 10(4), 425–437 (1997)
Rockafellar, R.T.: Convex Analysis, vol. 28. Princeton University Press, Princeton (1970)
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)
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)
Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, vol. 151, 2nd edn. Cambridge University Press, Cambridge (2014)
Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1986)
Sen, S., Sherali, H.D.: Nondifferentiable reverse convex programs and facetial convexity cuts via a disjunctive characterization. Math. Program. 37(2), 169–183 (1987)
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)
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)
Shor, N.Z.: Quadratic optimization problems. Sov. J. Comput. Syst. Sci. 25, 1–11 (1987)
Tardella, F.: Existence and sum decomposition of vertex polyhedral convex envelopes. Optim. Lett. 2(3), 363–375 (2008)
Tawarmalani, M., Richard, J.P.P., Xiong, C.: Explicit convex and concave envelopes through polyhedral subdivisions. Math. Program. 138(1–2), 531–577 (2013)
Tawarmalani, M., Sahinidis, N.V.: Convex extensions and envelopes of lower semi-continuous functions. Math. Program. 93(2), 247–263 (2002)
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)
Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)
Towle, E., Luedtke, J.: Intersection disjunctions for reverse convex sets. (2019). arXiv preprint arXiv:1901.02112
Tuy, H.: Concave programming under linear constraints. Sov. Math. 5, 1437–1440 (1964)
Vandenbussche, D., Nemhauser, G.: A branch-and-cut algorithm for nonconvex quadratic programs with box constraints. Math. Program. 102(3), 559–575 (2005)
Wolsey, L.A., Nemhauser, G.L.: Integer and Combinatorial Optimization. Wiley, New York (2014)
Xia, W., Vera, J.C., Zuluaga, L.F.: Globally solving nonconvex quadratic programs via linear integer programming techniques. INFORMS J. Comput. (2019)
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.
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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 (x, y) 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
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,
Solving Eq. (21) for \(\bar{r}\) with Eq. (22),
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
B Additional BoxQP experiments
See Table 5.
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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
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DOI: https://doi.org/10.1007/s10107-020-01484-3