# Deriving convex hulls through lifting and projection

- 166 Downloads

## Abstract

We consider convex hull descriptions for certain sets described by inequality constraints over a hypercube and propose a lifting-and-projection technique to construct them. The general procedure obtains the convex hulls as an intersection of semi-infinite families of linear inequalities, each derived using lifting techniques that are interpreted using convexification tools. We demonstrate that differentiability and concavity of certain perturbation functions help reduce the number of inequalities needed for this characterization. Each family of inequalities yields a few linear/nonlinear constraints fully characterized in the space of the original problem variables, when the projection problems are amenable to a closed-form solution. In particular, we illustrate the complete procedure by constructing the convex hulls of the subsets of a compact hypercube defined by the constraints \(x_{1}^{b_{1}} x_{2}^{b_{2}} \ge x_3\) and \(x_1 x_{2}^{b_{2}} \le x_3\), where \(b_1,b_2\ge 1\). As a consequence, we obtain a closed-form description of the convex hull of the bilinear equality \(x_1x_2=x_3\), in the presence of variable bounds, as an intersection of a few linear and nonlinear inequalities. This explicit characterization, hitherto unknown, can improve relaxation techniques for factorable functions, which utilize this equality to relax products of functions with known relaxations.

### Keywords

Convex hull Nonlinear knapsack sets Lifting Projection Bilinear sets Explicit descriptions### Mathematics Subject Classification

90C11 46N10 52A27 90C26 90C57## Notes

### Acknowledgements

The authors wish to thank the two referees and an associate editor for their suggestions, which have led to improvements in the presentation and structure of the paper.

### References

- 1.Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res.
**8**, 273–286 (1983)MathSciNetCrossRefMATHGoogle Scholar - 2.Anstreicher, K.M., Burer, S.: Computable representations for convex hulls of low-dimensional quadratic forms. Math. Program.
**124**, 33–43 (2010)MathSciNetCrossRefMATHGoogle Scholar - 3.Belotti, P., Miller, A., Namazifar, M.: Valid inequalities for sets defined by multilinear functions. In: Proceedings of the European Workshop on Mixed Integer Nonlinear Programming (2010)Google Scholar
- 4.Bonnans, J.F., Shapiro, A.: Optimization problems with perturbations: a guided tour. SIAM Rev.
**40**, 228–264 (1998)MathSciNetCrossRefMATHGoogle Scholar - 5.Costa, A., Liberti, L.: Relaxations of multilinear convex envelopes: dual is better than primal. In Experimental Algorithms, pp. 87–98. Springer, Berlin, Heidelberg (2012)Google Scholar
- 6.Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1952)MATHGoogle Scholar
- 7.Jach, M., Michaels, D., Weismantel, R.: The convex envelope of (\(n\)-1)-convex functions. SIAM J. Optim.
**19**, 1451–1466 (2008)MathSciNetCrossRefMATHGoogle Scholar - 8.Khajavirad, A., Sahinidis, N.V.: Convex envelopes generated from finitely many compact convex sets. Math. Program.
**137**, 371–408 (2013)MathSciNetCrossRefMATHGoogle Scholar - 9.Locatelli, M., Schoen, F.: On convex envelopes for bivariate functions over polytopes. Math. Program.
**144**, 65–91 (2014)MathSciNetCrossRefMATHGoogle Scholar - 10.Luedtke, J., Namazifar, M., Linderoth, J.: Some results on the strength of relaxations of multilinear functions. Math. Program.
**136**, 325–351 (2012)MathSciNetCrossRefMATHGoogle Scholar - 11.McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I-convex underestimating problems. Math. Program.
**10**, 147–175 (1976)CrossRefMATHGoogle Scholar - 12.Meyer, C.A., Floudas, C.A.: Trilinear monomials with mixed sign domains: facets of the convex and concave envelopes. J. Global Optim.
**29**, 125–155 (2004)MathSciNetCrossRefMATHGoogle Scholar - 13.Meyer, C.A., Floudas, C.A.: Convex envelopes for edge-concave functions. Math. Program.
**103**, 207–224 (2005)MathSciNetCrossRefMATHGoogle Scholar - 14.Richard, J.-P.P., Tawarmalani, M.: MIP lifting techniques for mixed integer nonlinear programs. In: Third Workshop on Mixed Integer Programming, Miami (2006)Google Scholar
- 15.Richard, J.-P.P., Tawarmalani, M.: Lifting inequalities: a framework for generating strong cuts for nonlinear programs. Math. Program.
**121**, 61–104 (2010)MathSciNetCrossRefMATHGoogle Scholar - 16.Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series. Princeton University Press, Princeton (1970)Google Scholar
- 17.Sherali, H.D.: Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets. Acta Math. Vietnam.
**22**, 245–270 (1997)MathSciNetMATHGoogle Scholar - 18.Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math.
**3**, 411–430 (1990)MathSciNetCrossRefMATHGoogle Scholar - 19.Tawarmalani, M., Richard, J.-P. P.: Decomposition techniques in convexification of inequalities. Krannert Working Paper (2015)Google Scholar
- 20.Tawarmalani, M., Sahinidis, N.V.: Semidefinite relaxations of fractional programs via novel techniques for constructing convex envelopes of nonlinear functions. J. Global Optim.
**20**, 137–158 (2001)MathSciNetCrossRefMATHGoogle Scholar - 21.Tawarmalani, M., Richard, J.-P.P., Chung, K.: Strong valid inequalities for orthogonal disjunctions and bilinear covering sets. Math. Program.
**124**, 481–512 (2010)MathSciNetCrossRefMATHGoogle Scholar - 22.Tawarmalani, M., Richard, J.-P.P., Xiong, C.: Explicit convex and concave envelopes through polyhedral subdivisions. Math. Program.
**138**, 531–577 (2013)MathSciNetCrossRefMATHGoogle Scholar