# Theorems of the Alternative for Conic Integer Programming

## TL;DR

This paper develops theorems of the alternative specifically for conic integer programming, extending foundational concepts like Farkas' Lemma to this more complex setting.

## Contribution

It introduces new theorems of the alternative for conic integer programming and provides a nested procedure to construct functions that characterize feasibility.

## Key findings

- Theorems of the alternative are extended to conic integer programming.
- A nested procedure for constructing feasibility-characterizing functions is proposed.
- The approach helps determine which statement in the theorem of the alternative holds.

## Abstract

Farkas' Lemma is a foundational result in linear programming, with implications in duality, optimality conditions, and stochastic and bilevel programming. Its generalizations are known as theorems of the alternative. There exist theorems of the alternative for integer programming and conic programming. We present theorems of the alternative for conic integer programming. We provide a nested procedure to construct a function that characterizes feasibility over right-hand sides and can determine which statement in a theorem of the alternative holds.