A Unified Theorem of the Alternative

TL;DR

This paper introduces a comprehensive theorem of the alternative that encompasses various types of relations such as equalities, inequalities, and dominance conditions, unifying multiple existing theorems into a single framework.

Contribution

It presents a unified theorem of the alternative that explicitly includes numerous relation types and nests 60 special cases, some previously stated separately.

Findings

Nests 60 special cases within a single theorem

Unifies multiple existing theorems of the alternative

Explicitly incorporates various relation types

Abstract

This note presents a unified theorem of the alternative that explicitly allows for any combination of equality, componentwise inequality, weak dominance, strict dominance, and nonnegativity relations. The theorem nests 60 special cases, some of which have been stated as separate theorems.