Lagrangian Relaxation for Integer Programming

It is a pleasure to write this commentary because it offers an opportunity to express my gratitude to several people who helped me in ways that turned out to be essential to the birth of [8]. They also had a good deal to do with shaping my early career and, consequently, much of what followed. The immediate event that triggered my interest in this topic occurred early in 1971 in connection with a consulting project I was doing for Hunt-Wesson Foods (now part of ConAgra Foods) with my colleague Glenn Graves. It was a distribution system design problem: how many distribution centers should there be and where, how should plant outputs flow through the DCs to customers, and related questions. We had figured out how to solve this large-scale MILP problem optimally via Benders Decomposition, a method that had been known for about a decade but had not yet seen practical application to our knowledge. This involved repeatedly solving a large 0-1 integer linear programming master problem in alternation with as many pure classical transportation subproblems as there were commodity classes. The master problem was challenging, and one day Glenn, who did all the implementation, came up with a new way to calculate conditional “penalties” to help decide which variable to branch on in our LP-based branch-and-bound approach. I regularly taught a doctoral course in those days that covered, inter alia, the main types of penalties used by branch-and-bound algorithms. But after studying the math that Glenn used to justify his, I didn’t see a connection to any of the penalties I knew about. I did, however, notice that Glenn made use of a Lagrangean term, and I was very familiar with Lagrangeans owing to my earlier work on solving discrete optimization problems via Lagrange multipliers [2] and on duality theory in nonlinear programming [6]. It often happens that a mathematical result can be