# Error bounds for monomial convexification in polynomial optimization

- 167 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.Adams, W., Gupte, A., Xu, Y.: An RLT approach for convexifying symmetric multilinear polynomials. Working paper (2017)Google Scholar
- 2.Al-Khayyal, F., Falk, J.: Jointly constrained biconvex programming. Math. Oper. Res.
**8**(2), 273–286 (1983)MathSciNetCrossRefzbMATHGoogle Scholar - 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)MathSciNetCrossRefzbMATHGoogle Scholar - 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)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Belotti, P., Miller, A.J., Namazifar, M.: Valid inequalities and convex hulls for multilinear functions. Electron. Notes Discrete Math.
**36**, 805–812 (2010)CrossRefzbMATHGoogle Scholar - 6.Benson, H.P.: Concave envelopes of monomial functions over rectangles. Naval Res. Logist. (NRL)
**51**(4), 467–476 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 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)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Buchheim, C., D’Ambrosio, C.: Monomial-wise optimal separable underestimators for mixed-integer polynomial optimization. J. Glob. Optim.
**67**(4), 759–786 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Buchheim, C., Michaels, D., Weismantel, R.: Integer programming subject to monomial constraints. SIAM J. Optim.
**20**(6), 3297–3311 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Crama, Y.: Concave extensions for nonlinear 0–1 maximization problems. Math. Program.
**61**(1–3), 53–60 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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)MathSciNetCrossRefzbMATHGoogle Scholar - 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)MathSciNetCrossRefzbMATHGoogle Scholar - 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)MathSciNetCrossRefzbMATHGoogle Scholar - 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.Del Pia, A., Khajavirad, A.: A polyhedral study of binary polynomial programs. Math. Oper. Res.
**42**, 389–410 (2016)MathSciNetzbMATHGoogle Scholar - 17.Dey, S.S., Gupte, A.: Analysis of MILP techniques for the pooling problem. Oper. Res.
**63**(2), 412–427 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim.
**11**(3), 796–817 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Lasserre, J.B.: An Introduction to Polynomial and Semi-algebraic Optimization, vol. 52. Cambridge University Press, Cambridge (2015)CrossRefzbMATHGoogle Scholar
- 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.Liberti, L., Pantelides, C.C.: Convex envelopes of monomials of odd degree. J. Glob. Optim.
**25**(2), 157–168 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Linderoth, J.: A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math. Program.
**103**(2), 251–282 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 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 zbMATHGoogle Scholar
- 24.Locatelli, M., Schoen, F.: On convex envelopes for bivariate functions over polytopes. Math. Program.
**144**(1–2), 65–91 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 25.Luedtke, J., Namazifar, M., Linderoth, J.: Some results on the strength of relaxations of multilinear functions. Math. Program.
**136**(2), 325–351 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 26.McCormick, G.: Computability of global solutions to factorable nonconvex programs: part I. Convex underestimating problems. Math. Program.
**10**(1), 147–175 (1976)CrossRefzbMATHGoogle Scholar - 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)MathSciNetCrossRefzbMATHGoogle Scholar - 28.Meyer, C., Floudas, C.: Convex envelopes for edge-concave functions. Math. Program.
**103**(2), 207–224 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 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)MathSciNetCrossRefzbMATHGoogle Scholar - 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)MathSciNetCrossRefzbMATHGoogle Scholar - 31.Pang, J.S.: Error bounds in mathematical programming. Math. Program.
**79**(1–3), 299–332 (1997)MathSciNetzbMATHGoogle Scholar - 32.Rikun, A.: A convex envelope formula for multilinear functions. J. Glob. Optim.
**10**(4), 425–437 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 33.Ryoo, H.S., Sahinidis, N.V.: Analysis of bounds for multilinear functions. J. Glob. Optim.
**19**(4), 403–424 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 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)MathSciNetzbMATHGoogle Scholar - 35.Sherali, H.D., Dalkiran, E., Liberti, L.: Reduced RLT representations for nonconvex polynomial programming problems. J. Glob. Optim.
**52**(3), 447–469 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 36.Speakman, E., Lee, J.: Quantifying double McCormick. Math. Oper. Res.
**42**(4), 1230–1253 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 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)MathSciNetCrossRefzbMATHGoogle Scholar - 38.Tawarmalani, M., Sahinidis, N.: Convex extensions and envelopes of lower semi-continuous functions. Math. Program.
**93**(2), 247–263 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 39.Tawarmalani, M., Sahinidis, N.: A polyhedral branch-and-cut approach to global optimization. Math. Program.
**103**(2), 225–249 (2005)MathSciNetCrossRefzbMATHGoogle Scholar