In April 1967 I and my family arrived into the US as fresh immigrants from behind the Iron Curtain. After a fruitful semester spent with George Dantzig’s group in Stanford, I started working at CMU. My debut in integer programming and entry ticket into Academia was the additive algorithm for 0-1 programming [B65], an implicit enumeration procedure based on logical tests akin to what today goes under the name of constraint propagation. As it used only additions and comparisons, it was easy to implement and was highly popular for a while. However, I was aware of its limitations and soon after I joined CMU I started investigating cutting plane procedures, trying to use for this purpose the tools of convex analysis: support functions and their level sets, maximal convex extensions, polarity, etc. During the five years starting in 1969, I proposed a number of procedures based on the central idea of intersection cuts [2] (numbered references are to those at the end of the paper, whereas mnemonicized ones are to the ones listed at the end of this introduction): Given any convex set S containing the LP optimum of a mixed integer program (MIP) but containing no feasible integer point in its interior, one can generate a valid cut by intersecting the boundary of S with the extreme rays of the cone defined by the optimal solution to the linear programming relaxation of the MIP and taking the hyperplane defined by the intersection points as the cut. The search for the most appropriate sets S in this role has led to the concept of outer polars and related constructs [3, 6]. In our days, the idea of intersection cuts has been revived in the form of cutting planes from convex sets with lattice-free interiors, and is the object of numerous investigations (e.g., [ALWW07], [BC07], [CM07], [DW07]). It was this line of research that has led to the idea of disjunctive programming, through a process outlined in section 1 of the paper below. Optimizing a function
[1]
Egon Balas,et al.
programming: Properties of the convex hull of feasible points *
,
1998
.
[2]
Gérard Cornuéjols,et al.
Minimal Valid Inequalities for Integer Constraints
,
2009,
Math. Oper. Res..
[3]
Egon Balas,et al.
A precise correspondence between lift-and-project cuts, simple disjunctive cuts, and mixed integer gomory cuts for 0-1 programming
,
2003,
Math. Program..
[4]
Laurence A. Wolsey,et al.
Lifting Integer Variables in Minimal Inequalities Corresponding to Lattice-Free Triangles
,
2008,
IPCO.
[5]
Egon Balas,et al.
The perfectly matchable subgraph polytope of a bipartite graph
,
1983,
Networks.
[6]
Egon Balas,et al.
The perfectly Matchable Subgraph Polytope of an arbitrary graph
,
1989,
Comb..
[7]
Laurence A. Wolsey,et al.
Compact formulations as a union of polyhedra
,
2005,
Math. Program..
[8]
Alper Atamtürk.
Strong Formulations of Robust Mixed 0–1 Programming
,
2006,
Math. Program..
[9]
Sanjay Mehrotra,et al.
A branch-and-cut method for 0-1 mixed convex programming
,
1999,
Math. Program..
[10]
Laurence A. Wolsey,et al.
Production Planning by Mixed Integer Programming
,
2010
.
[11]
Egon Balas,et al.
New Variants of Lift-and-Project Cut Generation from the LP Tableau: Open Source Implementation and Testing
,
2007,
IPCO.
[12]
Alexander Schrijver,et al.
Cones of Matrices and Set-Functions and 0-1 Optimization
,
1991,
SIAM J. Optim..
[13]
Bernard Cornet,et al.
Contributions to Operations Research and Economics
,
1990
.
[14]
Gérard Cornuéjols,et al.
On the facets of mixed integer programs with two integer variables and two constraints
,
2008,
Math. Program..