On ill-de fi nition conditions for “extended formulations” Moustapha Diaby OPIM Department; University of Connecticut; Storrs, CT 06268 moustapha.diaby@business.uconn.edu Mark H. Karwan Department of Industrial and Systems Engineering; SUNY at Buffalo; Amherst, NY 14260 mkarwan@buffalo.edu Abstract: The purpose of this note is to bring to attention and to make a contribution to the issue of defining/clarifying the scope of applicability of “extended formulations (EF’s)” theory. We show that there exists a limit to the scope within which the notion of an EF can be used to relate alternate models of a given optimization problem. Specifically, we show that the notion of an EF can become ill-defined (and thereby, lose its meaningfulness) when the polytopes being related are expressed in coordinate systems that are independent of each other. We briefly illustrate some of the ideas using the developments in Fiorini et al. (2011), in particular. Keywords: Linear Programming; Combinatorial Optimization; Traveling Salesman Problem; TSP; Com- putational Complexity, Extended Formulations. 1 Introduction There has been a renewed interest in Extended Formulations (EF’s) over the past 3 years (see Conforti et al. (2010), Vanderbeck and Wolsey (2010), Fiorini et al. (2011), and Kaibel (2011), for example). Despite the great importance of the EF paradigm in the analysis of linear programming (LP) and integer programming (IP) models of combinatorial optimization problems (COP’s), the clear definition of its scope of applicability has been largely an overlooked issue. The purpose of this note is to make a contribution towards addressing this issue. Specifically, we will show that the notion of an EF can become ill-defined (and thereby, lose its meaningfulness) when it is being used to relate polytopes involved in alternate abstractions of a given optimization problem. Because most of the papers on EF’s focus on the TSP specifically, we will center our discussion on the TSP. However, the substance of the paper is applicable for other NP-Complete problems. The plan of the paper is as follows. First, in section 2, we will review the basic definitions and notation, and show the error of any notion there may be, suggesting an “impossibility” of abstracting the TSP optimization problem over a polytope of polynomial size, by showing that TSP tours can be represented (by inference) independently of the TSP polytope. Then, we will introduce the notion of “polyhedron augmentation” in section 3, and use it (in section 4) to develop our results on the condition about EF’s becoming ill-defined. Finally, we will offer some concluding remarks in section 5. The general notation we will use is as follows. Notation 1 1. R : Set of real numbers; 1