The Theorems of the Alternative

Kathryn L. Balsman '91, Illinois Wesleyan University


In layman's terms, a Theorem of the Alternative is a theorem which states that given two conditions, one of the two conditions is true. It further states that if one of those conditions fails to be true, then the other condition must be true. Mathematically, this does not give a very clear picture of this group of theorems, however. Katta Murty defines a typical Theorem of the Alternative as one which shows that corresponding to any given system of linear constraints, system I, there is another associated system of linear constraints, system II, based on the same data, satisfying the property that one of the systems among I and II is feasible if and only if the other is infeasible. These theorems are not extremely well-known although they do have direct applications in the derivation of optimality conditions. The task at hand was to explore the Theorems of the Alternative as they are found in linear programming, projection theory, and linear algebra.