& \text{subject to} && A \mathbf{x} \leq \mathbf{b} \\
& \text{and} && \mathbf{x} \ge \mathbf{0}.
\end{align} </math>
whereHere the components of '''x''' representsare the vector of variables (to be determined), '''c''' and '''b''' are given [[vector space|vectors]] of (known)with <math>\mathbf{c}^T</math> indicating that the coefficients of '''c''' are used as a single-row matrix for the purpose of forming the matrix product), and ''A'' is a (known)given [[Matrix (mathematics)|matrix]]. ofThe coefficients,function andwhose value is to be maximized or minimized (<math>(\cdot)^mathbf x\mathrmmapsto\mathbf{c}^T\mathbf{x}</math> isin thethis [[matrix transpose]]. The expression to be maximized or minimizedcase) is called the [[objective function]] ('''c'''<sup>T</sup>'''x''' in this case). The inequalities ''A'''''x''' ≤ '''b''' and '''x''' ≥ '''0''' are the constraints which specify a [[convex polytope]] over which the objective function is to be optimized. In this context, two vectors are [[Comparability|comparable]] when they have the same dimensions. If every entry in the first is less-than or equal-to the corresponding entry in the second, then it can be said that the first vector is less-than or equal-to the second vector.
Linear programming can be applied to various fields of study. It is widely used in mathematics, and to a lesser extent in business, [[economics]], and for some engineering problems. Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing. It has proven useful in modeling diverse types of problems in [[automated planning and scheduling|planning]], [[routing]], [[scheduling (production processes)|scheduling]], [[assignment problem|assignment]], and design.
