Abstract
In this paper, we consider functions of the form \({\phi(x,y)=f(x)g(y)}\) over a box, where \({f(x), x\in {\mathbb R}}\) is a nonnegative monotone convex function with a power or an exponential form, and \({g(y), y\in {\mathbb R}^n}\) is a component-wise concave function which changes sign over the vertices of its domain. We derive closed-form expressions for convex envelopes of various functions in this category. We demonstrate via numerical examples that the proposed envelopes are significantly tighter than popular factorable programming relaxations.
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Al-Khayyal F.A., Falk J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8, 273–286 (1983)
Bao X., Sahinidis N.V., Tawarmalani M.: Multiterm polyhedral relaxations for nonconvex, quadratically-constrained quadratic programs. Optim. Methods Softw. 24, 485–504 (2009)
Bussieck M.R., Drud A.S., Meeraus A.: MINLPLib-a collection of test models for mixed-integer nonlinear programming. INFORMS J. Comput. 15, 114–119 (2003)
GLOBAL Library. http://www.gamsworld.org/global/globallib.htm
Hardy G.H., Littlewood J.E., Polya G.: Inequalities. Cambridge University Press, Cambridge (1952)
Hiriart-Urruty J.B., Lemaréchal C.: Fundamentals of Convex Analysis. Grundlehren Text Editions, Springer, Heidelberg (2001)
Jach M., Michaels D., Weismantel R.: The convex envelope of (n − 1)-convex functions. SIAM J. Optim. 19, 1451–1466 (2008)
Khajavirad, A., Sahinidis, N.V.: Convex envelopes generated from finitely many compact convex sets. Mathematical Programming (submitted)
McCormick G.P.: Computability of global solutions to factorable nonconvex programs: part I—Convex underestimating problems. Math. Program. 10, 147–175 (1976)
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)
Meyer C.A., Floudas C.A.: Convex envelopes for edge-concave functions. Math. Program. 103, 207–224 (2005)
Rikun A.D.: A convex envelope formula for multilinear functions. J. Global Optim. 10, 425–437 (1997)
Rockafellar R.T.: Convex Analysis. Princeton Mathematical Series. Princeton University Press, Princeton (1970)
Sahinidis N.V.: BARON: a general purpose global optimization software package. J. Global Optim. 8, 201–205 (1996)
Sherali H.D.: Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets. Acta Math. Vietnam. 22, 245–270 (1997)
Tardella F.: Existence and sum decomposition of vertex polyhedral convex envelopes. Optim. Lett. 2, 363–375 (2008)
Tawarmalani, M.: Inclusion certificates and simultaneous convexification of functions. Mathematical Programming (submitted)
Tawarmalani, M., Richard, J.P., Xiong, C.: Explicit convex and concave envelopes through polyhedral subdivisions. Mathematical Programming (submitted)
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)
Tawarmalani M., Sahinidis N.V.: Convex extensions and convex envelopes of l.s.c. functions. Math. Program. 93, 247–263 (2002)
Topkis D.M.: Supermodularity and complementarity. Princeton University Press, Princeton (1998)
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This research was supported in part by National Science Foundation award CMII-1030168.
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Khajavirad, A., Sahinidis, N.V. Convex envelopes of products of convex and component-wise concave functions. J Glob Optim 52, 391–409 (2012). https://doi.org/10.1007/s10898-011-9747-5
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DOI: https://doi.org/10.1007/s10898-011-9747-5