Error bounds for monomial convexification in polynomial optimization

Full Length Paper Series A
  • 57 Downloads

Abstract

Convex hulls of monomials have been widely studied in the literature, and monomial convexifications are implemented in global optimization software for relaxing polynomials. However, there has been no study of the error in the global optimum from such approaches. We give bounds on the worst-case error for convexifying a monomial over subsets of Open image in new window . This implies additive error bounds for relaxing a polynomial optimization problem by convexifying each monomial separately. Our main error bounds depend primarily on the degree of the monomial, making them easy to compute. Since monomial convexification studies depend on the bounds on the associated variables, in the second part, we conduct an error analysis for a multilinear monomial over two different types of box constraints. As part of this analysis, we also derive the convex hull of a multilinear monomial over Open image in new window .

Keywords

Polynomial optimization Monomial Multilinear Convex hull Error analysis Means inequality 

Mathematics Subject Classification

90C26 65G99 52A27 

Notes

Acknowledgements

The first author was supported in part by ONR Grant N00014-16-1-2168. The second author was supported in part by ONR Grant N00014-16-1-2725. We thank two referees whose meticulous reading helped us clarify some of the technical details.

References

  1. 1.
    Adams, W., Gupte, A., Xu, Y.: An RLT approach for convexifying symmetric multilinear polynomials. Working paper (2017)Google Scholar
  2. 2.
    Al-Khayyal, F., Falk, J.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273–286 (1983)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bao, X., Khajavirad, A., Sahinidis, N.V., Tawarmalani, M.: Global optimization of nonconvex problems with multilinear intermediates. Math. Program. Comput. 7(1), 1–37 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24(4), 597–634 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Belotti, P., Miller, A.J., Namazifar, M.: Valid inequalities and convex hulls for multilinear functions. Electron. Notes Discrete Math. 36, 805–812 (2010)CrossRefMATHGoogle Scholar
  6. 6.
    Benson, H.P.: Concave envelopes of monomial functions over rectangles. Naval Res. Logist. (NRL) 51(4), 467–476 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    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 (2017)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Buchheim, C., D’Ambrosio, C.: Monomial-wise optimal separable underestimators for mixed-integer polynomial optimization. J. Glob. Optim. 67(4), 759–786 (2017)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Buchheim, C., Michaels, D., Weismantel, R.: Integer programming subject to monomial constraints. SIAM J. Optim. 20(6), 3297–3311 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Crama, Y.: Concave extensions for nonlinear 0–1 maximization problems. Math. Program. 61(1–3), 53–60 (1993)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Crama, Y., Rodríguez-Heck, E.: A class of valid inequalities for multilinear 0–1 optimization problems. Discrete Optim. 25, 28–47 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dalkiran, E., Sherali, H.D.: RLT-POS: reformulation-linearization technique-based optimization software for solving polynomial programming problems. Math. Program. Comput. 8, 1–39 (2016)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    De Klerk, E., Laurent, M.: Error bounds for some semidefinite programming approaches to polynomial minimization on the hypercube. SIAM J. Optim. 20(6), 3104–3120 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    De Klerk, E., Laurent, M., Sun, Z.: An error analysis for polynomial optimization over the simplex based on the multivariate hypergeometric distribution. SIAM J. Optim. 25(3), 1498–1514 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    De Klerk, E., Laurent, M., Sun, Z.: Convergence analysis for Lasserres measure-based hierarchy of upper bounds for polynomial optimization. Math. Program. 162, 1–30 (2016)MathSciNetGoogle Scholar
  16. 16.
    Del Pia, A., Khajavirad, A.: A polyhedral study of binary polynomial programs. Math. Oper. Res. 42, 389–410 (2016)MathSciNetMATHGoogle Scholar
  17. 17.
    Dey, S.S., Gupte, A.: Analysis of MILP techniques for the pooling problem. Oper. Res. 63(2), 412–427 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lasserre, J.B.: An Introduction to Polynomial and Semi-algebraic Optimization, vol. 52. Cambridge University Press, Cambridge (2015)CrossRefMATHGoogle Scholar
  20. 20.
    Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Emerging Applications of Algebraic Geometry, pp. 157–270. Springer (2009)Google Scholar
  21. 21.
    Liberti, L., Pantelides, C.C.: Convex envelopes of monomials of odd degree. J. Glob. Optim. 25(2), 157–168 (2003)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Linderoth, J.: A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math. Program. 103(2), 251–282 (2005)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Locatelli, M.: Polyhedral subdivisions and functional forms for the convex envelopes of bilinear, fractional and other bivariate functions over general polytopes. J. Glob. Optim. Online First (2016).  https://doi.org/10.1007/s10898-016-0418-4 MATHGoogle Scholar
  24. 24.
    Locatelli, M., Schoen, F.: On convex envelopes for bivariate functions over polytopes. Math. Program. 144(1–2), 65–91 (2014)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Luedtke, J., Namazifar, M., Linderoth, J.: Some results on the strength of relaxations of multilinear functions. Math. Program. 136(2), 325–351 (2012)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    McCormick, G.: Computability of global solutions to factorable nonconvex programs: part I. Convex underestimating problems. Math. Program. 10(1), 147–175 (1976)CrossRefMATHGoogle Scholar
  27. 27.
    Meyer, C., Floudas, C.: Trilinear monomials with mixed sign domains: facets of the convex and concave envelopes. J. Glob. Optim. 29(2), 125–155 (2004)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Meyer, C., Floudas, C.: Convex envelopes for edge-concave functions. Math. Program. 103(2), 207–224 (2005)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Misener, R., Floudas, C.A.: Antigone: algorithms for continuous/integer global optimization of nonlinear equations. J. Glob. Optim. 59(2–3), 503–526 (2014)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    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 Softw. 30(1), 215–249 (2015)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Pang, J.S.: Error bounds in mathematical programming. Math. Program. 79(1–3), 299–332 (1997)MathSciNetMATHGoogle Scholar
  32. 32.
    Rikun, A.: A convex envelope formula for multilinear functions. J. Glob. Optim. 10(4), 425–437 (1997)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Ryoo, H.S., Sahinidis, N.V.: Analysis of bounds for multilinear functions. J. Glob. Optim. 19(4), 403–424 (2001)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Sherali, H.: Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets. Acta Math. Vietnam. 22(1), 245–270 (1997)MathSciNetMATHGoogle Scholar
  35. 35.
    Sherali, H.D., Dalkiran, E., Liberti, L.: Reduced RLT representations for nonconvex polynomial programming problems. J. Glob. Optim. 52(3), 447–469 (2012)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Speakman, E., Lee, J.: Quantifying double McCormick. Math. Oper. Res. 42(4), 1230–1253 (2017)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    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
  38. 38.
    Tawarmalani, M., Sahinidis, N.: Convex extensions and envelopes of lower semi-continuous functions. Math. Program. 93(2), 247–263 (2002)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Tawarmalani, M., Sahinidis, N.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mathematical SciencesClemson UniversityClemsonUSA

Personalised recommendations