A new theorem of the alternative – a location approach

Abstract

This paper provides a new method to obtain theorems of the alternative, using a location approach. This consists of associating a location problem with the original system of linear inequalities. The paper introduces a dual optimization problem and strong duality is established. From the dual problem, there is derived a second system of equations and inequalities and for both systems the duality and solvability properties provide a conclusion of the general theorem of the alternative. The assigned location problem includes weighting factors and powers of the norms. These weighting factors and exponents may be considered as parameters appearing in the theorem of the alternative, which therefore represents a whole class of theorems of the alternative. The presented theorem of the alternative may be considered as a generalization of the well-known Gale's theorem of the alternative for linear inequalities.