Mathematical Methods of Operations Research https://doi.org/10.1007/s00186-020-00723-9 ORIGINAL ARTICLE On the facet defining inequalities of the mixed-integer bilinear covering set Hamidur Rahman 1 · Ashutosh Mahajan 1 Received: 21 December 2018 / Revised: 29 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020 Abstract We study the facet defining inequalities of the convex hull of a mixed-integer bilinear covering arising in trim-loss (or cutting stock) problem under the framework of dis- junctive cuts. We show that all of them can be derived using a disjunctive procedure. Some of these are split cuts of rank one for a convex mixed-integer relaxation of the covering set, while others have rank at least two. For certain linear objective functions, the rank-one split cuts are shown to be sufficient for finding the optimal value over the convex hull of the covering set. A relaxation of the trim-loss problem has this property, and our computational results show that these rank-one inequalities find the lower bound quickly. Keywords Mixed-integer programming · Global optimization · Convex hull · Disjunctive cut · Split cut · Split-rank 1 Introduction We study the facet defining inequalities of the convex hull of the mixed-integer bilinear covering set S =  (x , y ) ∈ Z n + × R n + : n  i =1 x i y i ≥ r  , where r > 0. This set appears in real life applications like trim loss (or cutting stock) problem (Harjunkoski et al. 1998; Vanderbeck 2000); see Sect. 9 for more details. The B Ashutosh Mahajan amahajan@iitb.ac.in Hamidur Rahman iamhrahmankck@gmail.com 1 Industrial Engineering and Operations Research, Indian Institute of Technology Bombay, Mumbai 400 076, India 123