# Duality Theory

• Robert J. Vanderbei
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 37)

## Abstract

Associated with every linear program is another called its dual. The dual of this dual linear program is the original linear program (which is then referred to as the primal linear program). Hence, linear programs come in primal/dual pairs. It turns out that every feasible solution for one of these two linear programs gives a bound on the optimal objective function value for the other. These ideas are important and form a subject called duality theory, which is the topic we shall study in this chapter.

## Keywords

Dual Problem Linear Programming Problem Duality Theory Simplex Method Primal Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

1. The idea behind the strong duality theorem can be traced back to conversations between G.B. Dantzig and J. von Neumann in the fall of 1947, but an explicit statement did not surface until the paper of Gale et al. (1951). The term primal problem was coined by G.B. Dantzig’s father, T. Dantzig. The dual simplex method was first proposed by Lemke (1954).Google Scholar
2. The solution to Exercise 5.13 (which is left to the reader to supply) suggests that a random linear programming problem is infeasible with probability 1/4, unbounded with probability 1/4, and has an optimal solution with probability 1/2.Google Scholar