Mixed Integer Linear Programming

A Mixed Integer Linear Programming problem has the following form:

Where f is a n x 1 vector containing the linear objective function, which is subject to the following constraints:

Linear Inequalities*

A is a m x n sparse matrix, b is a m x 1 vector

Linear Equalities*

Aeq is a k x n sparse matrix, beq is a k x 1 vector

Decision Variable Bounds

lb and ub are n x 1 vectors, where -inf or inf indicate an unbounded lower or upper bound, respectively

Integer Constraints

x_i are decision variables which must be a integer number (...-2, -1, 0, 1, 2...)

Binary Constraints

x_j are decision variables which must be a binary number (0,1), where i ≠ j.

The goal is to minimize the objective function by selecting a value of x that also satisfies all constraints.