Mixing MIR inequalities with two divisible coefficients

• Published in: Mathematical programming Vol. 123; no. 2; pp. 451 - 483
• Main Authors: Constantino, Miguel, Miller, Andrew J., Van Vyve, Mathieu
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.06.2010; Springer Nature B.V
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Costs; Full Length Paper; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Production planning; Theoretical; 90C11
• ISSN: 0025-5610; 1436-4646; 1436-4646
• DOI: 10.1007/s10107-009-0266-9

This article is a polyhedral study of a generalization of the mixing set where two different, divisible coefficients are allowed for the integral variables. Our results generalize earlier work on mixed integer rounding, mixing, and extensions. They also directly apply to applications such as production planning problems involving lower bounds or start-ups on production, when these are modeled as mixed-integer linear programs. We define a new class of valid inequalities and give two proofs that they suffice to describe the convex hull of this mixed-integer set. We give a characterization of each of the maximal faces of the convex hull, as well as a closed form description of its extreme points and rays, and show how to separate over this set in . Finally, we give several extended formulations of polynomial size, and study conditions under which adding certain simple constraints on the integer variables preserves our main result.