Linear Programming

Synonyms

LP

Definition

A linear programming problem (also termed linear program) is an optimization problem to minimize or maximize a linear objective function subject to linear equality/inequality constraints. Linear programming, often abbreviated as LP, is a methodology initiated by G. Dantzig, J. von Neumann, L. V. Kantorovich, and others in the 1940s. [1–6, 10, 11, 15, 17] It includes:

• Modeling techniques to formulate real-world problems into linear programs,

• Theory about the mathematical structure of linear programs,

• Algorithms for numerically solving linear programs.

Standard Form

For theoretical treatment and software development, it is convenient to use a specific form to describe linear programs. Often adopted for use is:

$$ \begin{array}{@{}ll@{}} \mbox{Minimize } & c^{\top} x \\ \mbox{subject to} & Ax = b \\ & x \geq {\bf 0}, \\ \end{array} $$

(1)

called the standard form, although the terminology varies in the literature. It should be clear that:

• An instance of the...