An extension of the fundamental theorem of linear programming

In 1947 George Dantzig developed the Simplex Algorithm for linear programming, and in doing so became known as The Father of Linear Programming. The invention of the Simplex Algorithm has been called ‘one of the most important discoveries of the 20th century’, and linear programming techniques have proven useful in numerous fields of study, As such, topics in linear optimization are taught in a variety of disciplines. The finite convergence of the simplex algorithm hinges on a result stating that every linear program with an optimal solution has a basic optimal solution; a result known as the fundamental theorem of linear programming. We develop an analog to the fundamental theorem, and the perspective from which we view the problem allows a much greater class of function. Indeed, not only do we relinquish the assumption of linearity, but we also do not assume the functions under consideration are continuous. Our new result implies the fundamental theorem of linear programming.