Duality Theory and Optimality Conditions for LPs

Abstract

Associated with every linear programming problem, there is another linear program called its dual, involving a different set of variables, but sharing the same data. When referring to the dual problem of an LP, the original LP is called the primal or the primal problem. Together, the two problems are referred to as a primal, dual pair of linear programs. The names primal, dual for the two problems are coined by Tobias Dantzig, father of George Dantzig, around 1955 in conversations with his son.

A duality type result for systems of linear equations only (no inequalities) is the theorem of alternatives for systems of linear equations (Theorem 1.1 in Sect. 1.2); it has been known for a long time (by the eighteenth century or even earlier), but similar results for systems of linear constraints including linear inequalities were unknown until recently. These important duality-type results for systems of linear constraints including inequalities known as either/or theorems or theorems of alternatives started appearing in published literature beginning in mid-nineteenth century.