Journal of Global Optimization

, Volume 60, Issue 2, pp 217–238 | Cite as

Extended formulations for convex envelopes



In this work we derive explicit descriptions for the convex envelope of nonlinear functions that are component-wise concave on a subset of the variables and convex on the other variables. These functions account for more than 30 % of all nonlinearities in common benchmark libraries. To overcome the combinatorial difficulties in deriving the convex envelope description given by the component-wise concave part of the functions, we consider an extended formulation of the convex envelope based on the Reformulation–Linearization-Technique introduced by Sherali and Adams (SIAM J Discret Math 3(3):411–430, 1990). Computational results are reported showing that the extended formulation strategy is a useful tool in global optimization.


Convex envelope Edge-concave functions Extended formulation Reformulation–Linearization-Technique Simultaneous convexification 



This work is part of the Collaborative Research Centre “Integrated Chemical Processes in Liquid Multiphase Systems” (CRC/Transregio 63 “InPROMPT”) funded by the German Research Foundation (DFG). Main parts of this work have been finished while the second author was at the Institute for Operations Research at ETH Zurich and financially supported by DFG through the CRC/Transregio 63. The authors thank the DFG for its financial support. We would like to thank Jon Lee for providing the Lavor test instances used in this paper. We are grateful to Stefan Vigerske for his continued support for SCIP.


  1. 1.
    Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1(1), 1–41 (2009)CrossRefGoogle Scholar
  2. 2.
    Adams, W.P., Sherali, H.D.: A hierarchy of relaxations leading to the convex hull representation for general discrete optimization problems. Ann. Oper. Res. 140(1), 21–47 (2005)CrossRefGoogle Scholar
  3. 3.
    Anstreicher, K.M., Burer, S.: Computable representations for convex hulls of low-dimensional quadratic forms. Math. Program. 124, 33–43 (2010)CrossRefGoogle Scholar
  4. 4.
    Ballerstein, M., Michaels, D.: Convex underestimation of edge-concave functions by a simultaneous convexification with multi-linear monomials. In: Alonse, D., Hansen, P., Rocha, C. (eds.) Proceedings of the Global Optimization Workshop, pp. 35–38 (2012). Available at
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    Barber, C.B., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22(4), 469–483 (1996)CrossRefGoogle Scholar
  7. 7.
    Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24(4–5), 597–634 (special issue: Global Optimization) (2009)Google Scholar
  8. 8.
    Burer, S., Letchford, A.: On non-convex quadratic programming with box constraints. SIAM J. Optim. 20, 1073–1089 (2009)CrossRefGoogle Scholar
  9. 9.
    Bussieck, M.R., Drud, A.S., Meeraus, A.: MINLPLib—a collection of test models for mixed-integer nonlinear programming. INFORMS J. Comput. 15(1), 114–119 (2003)CrossRefGoogle Scholar
  10. 10.
    Cafieri, S., Lee, J., Liberti, L.: On convex relaxations of quadrilinear terms. J. Glob. Optim. 47(4), 661–685 (2010)CrossRefGoogle Scholar
  11. 11.
  12. 12.
    Huggins, P., Sturmfels, B., Yu, J., Yuster, D.: The hyperdeterminant and triangulations of the 4-cube. Math. Comput. 77, 1653–1679 (2008)CrossRefGoogle Scholar
  13. 13.
  14. 14.
    Jach, M., Michaels, D., Weismantel, R.: The convex envelope of (\(n\)-1)-convex functions. SIAM J. Optim. 19(3), 1451–1466 (2008)CrossRefGoogle Scholar
  15. 15.
    Khajavirad, A., Sahidinidis, N.V.: Convex envelopes of products of convex and component-wise concave functions. J. Glob. Optim. 52, 391–409 (2012)CrossRefGoogle Scholar
  16. 16.
    Khajavirad, A., Sahinidis, N.V.: Convex envelopes generated from finitely many compact convex sets. Math. Program. Ser. A 137, 371–408 (2013)CrossRefGoogle Scholar
  17. 17.
    Laurent, M.: A comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre relaxations for 0–1 programming. Math. Oper. Res. 28(3), 470–496 (2003)CrossRefGoogle Scholar
  18. 18.
    Lavor, C.: On generating instances for the molecular distance geometry problem. In: Liberti, L., Maculan, N. (eds.) Global Optimization. From Theory to Implementation, pp. 405–414. Springer, Berlin (2006)Google Scholar
  19. 19.
    Lavor, C., Liberti, L., Maculan, N.: Molecular distance geometry problem. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, 2nd edn, pp. 2304–2311. Springer, Berlin (2009)Google Scholar
  20. 20.
    Locatelli, M.: Convex Envelopes for Quadratic and Polynomial Functions Over Polytopes (manuscript, 11 Mar 2010). Available at (2010)
  21. 21.
    Locatelli, M., Schoen, F.: On the Convex Envelopes and Underestimators For Bivariate Functions (manuscript, 17 Nov 2009). Available at (2009)
  22. 22.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs. I: convex underestimating problems. Math. Program. 10, 147–175 (1976)CrossRefGoogle Scholar
  23. 23.
    Meyer, C.A., Floudas, C.A.: Convex envelopes for edge-concave functions. Math. Program. 103, 207–224 (2005)CrossRefGoogle Scholar
  24. 24.
    Rockafellar, R.T.: Convex Analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1970)Google Scholar
  25. 25.
    SCIP: Solving Constraint Integer Programs (2009). Available at
  26. 26.
    Sherali, H.D.: Convex envelopes of multilinear functions over a unit hypercube and over special sets. Acta Math. Vietnam. 22(1), 245–270 (1997)Google Scholar
  27. 27.
    Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero–one programming problems. SIAM J. Discret. Math. 3(3), 411–430 (1990)CrossRefGoogle Scholar
  28. 28.
    Sherali, H.D., Adams, W.P.: A hierarchy of relaxations and convex hull characterizations for mixed-integer zero–one programming problems. Discret. Appl. Math. 52(1), 83–106 (1994)CrossRefGoogle Scholar
  29. 29.
    Sherali, H.D., Dalkiran, E., Desai, J.: Enhancing RLT-based relaxations for polynomial programming problems via a new class of v-semidefinite cuts. Comput. Optim. Appl. 52, 483–506 (2012)CrossRefGoogle Scholar
  30. 30.
    Sherali, H.D., Dalkiran, E., Liberti, L.: Reduced RLT representations for nonconvex polynomial programming problems. J. Glob. Optim. 52(3), 447–469 (2012)CrossRefGoogle Scholar
  31. 31.
    Sherali, H.D., Tuncbilek, C.H.: A global optimization algorithm for polynomial programming problems using a reformulation–linearization technique. J. Glob. Optim. 2, 101–112 (1992)CrossRefGoogle Scholar
  32. 32.
    Tardella, F.: On the existence of polyedral convex envelopes. In: Floudas, C.A., Pardalos, P.M. (eds.) Frontiers in Global Optimization, pp. 563–573. Kluwer, Dordrecht (2003)Google Scholar
  33. 33.
    Tardella, F.: Existence and sum decomposition of vertex polyhedral convex envelopes. Optim. Lett. 2, 363–375 (2008)CrossRefGoogle Scholar
  34. 34.
    Tawarmalani, M.: Inclusion Certificates and Simultaneous Convexification of Functions (manuscript, 5 Sept 2010). Available at (2010)
  35. 35.
    Tawarmalani, M., Richard, J.P.P., Xiong, C.: Explicit convex and concave envelopes through polyhedral subdivisions. Math. Program. Ser. A 138(1–2), 531–577 (2013)CrossRefGoogle Scholar
  36. 36.
    Tawarmalani, M., Sahinidis, N.V.: Semidefinite relaxations of fractional programs via novel convexification techniques. J. Glob. Optim. 20, 137–158 (2001)CrossRefGoogle Scholar
  37. 37.
    Tawarmalani, M., Sahinidis, N.V.: Convex extensions and envelopes of lower semi-continuous functions. Math. Program. 93, 247–263 (2002)CrossRefGoogle Scholar
  38. 38.
    Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99, 563–591 (2004)CrossRefGoogle Scholar
  39. 39.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)CrossRefGoogle Scholar
  40. 40.
    Wolfram Research: Mathematica. Wolfram Research, Champaign (2008)Google Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Eidgenössische Technische Hochschule ZürichInstitut für Operations ResearchZurichSwitzerland
  2. 2.Fakultät für MathematikTechnische Universität DortmundDortmundGermany

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