Disjunctive programming and relaxations of polyhedra

A hierarchy of relaxations for nonlinear convex generalized disjunctive programming

We propose a framework to generate alternative mixed-integer nonlinear programming formulations for disjunctive convex programs that lead to stronger relaxations. We extend the concept of “basic steps” defined for disjunctive linear programs to the nonlinear case. A basic step is an operation that takes a disjunctive set to another with fewer number of conjuncts. We show that the strength of th...

Disjunctive programming and a hierarchy of relaxations for discrete optimization problems

We discuss a new conceptual framework for the convexification of discrete optimization problems, and a general technique for obtaining approximations to the convex hull of the feasible set. The concepts come from disjunctive programming and the key tool is a description of the convex hull of a union of polyhedra in terms of a higherdimensional polyhedron. Although this description was known for...

Disjunctive Programming

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 ...

Copositive programming: separation and relaxations

A large portion of research in science and engineering, as well as in business, concerns one similar problem: how to make things “better”? Once properly modeled (often a highly nontrivial task), this kind of question can be approached via a mathematical optimization problem. An optimal solution to a mathematical optimization problem, when interpreted properly, might correspond to new knowledge,...

Stronger Linear Programming Relaxations