Skip to main content
SpringerLink
Log in

On convex relaxations of quadrilinear terms

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

The best known method to find exact or at least ε-approximate solutions to polynomial-programming problems is the spatial Branch-and-Bound algorithm, which rests on computing lower bounds to the value of the objective function to be minimized on each region that it explores. These lower bounds are often computed by solving convex relaxations of the original program. Although convex envelopes are explicitly known (via linear inequalities) for bilinear and trilinear terms on arbitrary boxes, such a description is unknown, in general, for multilinear terms of higher order. In this paper, we study convex relaxations of quadrilinear terms. We exploit associativity to rewrite such terms as products of bilinear and trilinear terms. Using a general technique, we formally establish the intuitive fact that any relaxation for k-linear terms that employs a successive use of relaxing bilinear terms (via the bilinear convex envelope) can be improved by employing instead a relaxation of a trilinear term (via the trilinear convex envelope). We present a computational analysis which helps establish which relaxations are strictly tighter, and we apply our findings to two well-studied applications: the Molecular Distance Geometry Problem and the Hartree–Fock Problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adjiman, C.S.: Global optimization techniques for process systems engineering. PhD thesis, Princeton University (1998)

  2. Adjiman C.S., Dallwig S., Floudas C.A., Neumaier A.: A global optimization method, αBB, for general twice-differentiable constrained NLPs: I. Theoretical advances. Comput. Chem. Eng. 22(9), 1137–1158 (1998)

    Article  Google Scholar 

  3. Al-Khayyal F.A., Falk J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273–286 (1983)

    Article  Google Scholar 

  4. Avis, D.: lrs. cgm.cs.mcgill.ca/~avis/C/lrs.html

  5. 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)

    Article  Google Scholar 

  6. Crippen G.M., Havel T.F.: Distance Geometry and Molecular Conformation. Wiley, New York (1988)

    Google Scholar 

  7. Floudas, C.: Personal communication (2007)

  8. Fourer R., Gay D.: The AMPL Book. Duxbury Press, Pacific Grove (2002)

    Google Scholar 

  9. Fukuda, K.: cdd. www.ifor.math.ethz.ch/~fukuda/cdd_home/cdd.html

  10. Gill, P.E.: User’s guide for SNOPT version 7. Systems Optimization Laboratory, Stanford University, California (2006)

  11. Gounaris C.E., Floudas C.A.: Tight convex underestimators for C 2-continuous problems: I. Multivariate functions. J. Glob. Optim. 42(1), 69–89 (2008)

    Article  Google Scholar 

  12. Gounaris C.E., Floudas C.A.: Tight convex underestimators for C 2-continuous problems: I. Univariate functions. J. Glob. Optim. 42(1), 51–67 (2008)

    Article  Google Scholar 

  13. ILOG: ILOG CPLEX 11.0 User’s Manual. (ILOG S.A. Gentilly, France 2008)

  14. Jach M., Michaels D., Weismantel R.: The convex envelope of (n-1)-convex functions. SIAM J. Optim. 19(3), 1451–1466 (2008)

    Article  Google Scholar 

  15. Lavor, C.: On generating instances for the molecular distance geometry problem. In Liberti and Maculan [22], pp 405–414

  16. Lavor C., Liberti L., Maculan N.: Computational experience with the molecular distance geometry problem. In: Pintér, J. (eds) Global Optimization: Scientific and Engineering Case Studies, pp. 213–225. Springer, Berlin (2006)

    Google Scholar 

  17. Lavor C., Liberti L., Maculan N.: Molecular distance geometry problem. In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization, pp. 2305–2311. Springer, New York (2008)

    Google Scholar 

  18. Lavor C., Liberti L., Maculan N., Chaer Nascimento M.A.: Solving Hartree–Fock systems with global optimization metohds. Europhys. Lett. 5(77), 50006p1–50006p5 (2007)

    Google Scholar 

  19. Lee J., Morris W.D. Jr: Geometric comparison of combinatorial polytopes. Discrete Appl. Math. 55(2), 163–182 (1994)

    Article  Google Scholar 

  20. Liberti L. : Comparison of convex relaxations for monomials of odd degree. In: Tseveendorj, I., Pardalos, P.M., Enkhbat, R. (eds) Optimization and Optimal Control, World Scientific, Singapore (2003)

    Google Scholar 

  21. Liberti, L.: Writing global optimization software. In Liberti and Maculan [22], pp 211–262 (2006)

  22. Liberti, L., Maculan, N. (eds.): Global Optimization: From Theory to Implementation. Springer, Berlin (2006)

    Google Scholar 

  23. Liberti L., Pantelides C.C.: Convex envelopes of monomials of odd degree. J. Glob. Optim. 25, 157–168 (2003)

    Article  Google Scholar 

  24. Liberti L., Cafieri S., Tarissan F.: Reformulations in mathematical programming: a computational approach. In: Abraham, A., Hassanien, A.-E., Siarry, P., Engelbrecht, A. (eds) Foundations on Computational Intelligence, vol. 3, Studies in Computational Intelligence volume 203, pp. 153–234. Springer, Berlin (2009)

    Chapter  Google Scholar 

  25. Liberti L., Lavor C., Chaer Nascimento M.A., Maculan N.: Reformulation in mathematical programming: an application to quantum chemistry. Discrete Appl. Math. 157(6), 1309–1318 (2009)

    Article  Google Scholar 

  26. Liberti L., Lavor C., Maculan N., Marinelli F.: Double variable neighbourhood search with smoothing for the molecular distance geometry problem. J. Glob. Optim. 43, 207–218 (2009)

    Article  Google Scholar 

  27. Liberti, L., Tsiakis, P., Keeping, B., Pantelides, C.C.: \({oo\mathcal {OPS}}\) . Centre for Process Systems Engineering, Chemical Engineering Department, Imperial College, London, UK (2001)

  28. McCormick G.P.: Computability of global solutions to factorable nonconvex programs: Part I—Convex underestimating problems. Math. program. 10, 146–175 (1976)

    Article  Google Scholar 

  29. Meyer C.A., Floudas C.A.: Trilinear monomials with positive or negative domains: Facets of the convex and concave envelopes. In: Floudas, C.A., Pardalos, P.M. (eds) Frontiers in Global Optimization, pp. 327–352. Kluwer, Amsterdam (2003)

    Google Scholar 

  30. Meyer C.A., Floudas C.A.: Trilinear monomials with mixed sign domains: facets of the convex and concave envelopes. J. Glob. Optim. 29, 125–155 (2004)

    Article  Google Scholar 

  31. Mosses P. : Denotational semantics. In: Leeuwen, J. (eds) Handbook of Theoretical Computer Science, vol. B, pp. 575–631. Elsevier, Amsterdam (1990)

    Google Scholar 

  32. Neumaier A.: Molecular modeling of proteins and mathematical prediction of protein structure. SIAM Rev 39, 407–460 (1997)

    Article  Google Scholar 

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

    Article  Google Scholar 

  34. Ryoo H.S., Sahinidis N.V.: A branch-and-reduce approach to global optimization. J. Glob. Optim. 8(2), 107–138 (1996)

    Article  Google Scholar 

  35. Ryoo H.S., Sahinidis N.V.: Analysis of bounds for multilinear functions. J. Glob. Optim. 19, 403–424 (2001)

    Article  Google Scholar 

  36. Sahinidis, N.V., Tawarmalani, M.: BARON 7.2.5: Global Optimization of Mixed-Integer Nonlinear Programs, User’s Manual (2005)

  37. Smith, E.M.B.: on the optimal design of continuous processes. PhD thesis, Imperial College of Science, Technology and Medicine, University of London, October (1996)

  38. Smith E.M.B., Pantelides C.C.: A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 23, 457–478 (1999)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  41. Tawarmalani M., Sahinidis N.V.: Global optimization of mixed integer nonlinear programs: a theoretical and computational study. Math. program. 99, 563–591 (2004)

    Article  Google Scholar 

  42. Yoon, J.-M., Gad, Y., Wu, Z.: Mathematical modeling of protein structure using distance geometry. Technical Report TR00-24, Dept. Comput. Applied Maths, Rice University, Houston (2000)

Download references

Author information

Authors and Affiliations

  1. LIX, École Polytechnique, 91128, Palaiseau, France

    Sonia Cafieri & Leo Liberti

  2. Department of Mathematical Sciences, IBM T.J. Watson Research Center, PO Box 218, Yorktown Heights, NY, 10598, USA

    Jon Lee

Authors
  1. Sonia Cafieri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jon Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Leo Liberti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sonia Cafieri.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cafieri, S., Lee, J. & Liberti, L. On convex relaxations of quadrilinear terms. J Glob Optim 47, 661–685 (2010). https://doi.org/10.1007/s10898-009-9484-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-009-9484-1

Keywords

Search

Navigation