Extended formulations for convex envelopes

Abstract

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.

Extended formulations for convex hulls of some bilinear functions

“…Our aim is to find a polytope P such that X (f ) = π[f ](P ), so that this P is a compact extended formulation. Observe that There are constructive methods for deriving extended formulations of X (f ) with exponentially many variables and facet-defining inequalities, such as using the extreme point characterization in (1) or the nontrivial approach of using the Sherali-Adams hierarchy [SA90] which can also be applied to more general nonlinear functions [BM14]. We restrict our attention to finding extended formulations in the quadratic space of (x, y) variables.…”

Limits to the scope of applicability of extended formulations theory for LP models of combinatorial optimisation problems

“…Work in the second group include Gouveia et al (2013), Ballerstein and Michaels (2014), Bärmann et al (2014), Buchanan and Butenko (2014), Godinho et al (2014), Lancia and Serafini (2014) and Leggieri et al (2014). Examples of work in the third group include Kaibel (2011), Fiorini et al (2011, 2012a, 2012b), Faenza et al (2012, Gillis and Glineur (2012) and Kaibel and Walter (2014).…”