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, 2, 3, 4, 5, 6, 7, 8, 9, 10]. 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

Background

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

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

(1)

called the standard form, although the terminology varies in the literature. It...