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

The purpose of this paper is to bring to attention and to make a contribution to the issue of defining/clarifying the scope of applicability of extended formulations (EFs) theory. Specifically, we show that EFs theory is not valid for relating the sizes of descriptions of polytopes when the sets of the descriptive variables for those polytopes are disjoint, and that new definitions of the notion of 'projection' upon which some of the recent extended formulations works [such as Kaibel (2011), Fiorini et al. (2011, 2012a, 2012b, 2013), Faenza et al. (2012), Gillis and Glineur (2012) and Kaibel and Walter (2014), for example] have been based can cause those works to over-reach in their conclusions.