Overview of Linear Programming

We often use matrix form to express a linear programming problem.

\begin{array}{rllllllll} \min & f(\mathbf{x}) = \mathbf{c}^{\top} \mathbf{x} \\ \text{s.t.} & \mathbf{A}\mathbf{x}=\mathbf{b} \\ & \mathbf{x} \ge 0 \\ \text{where} & \mathbf{x} = \begin{bmatrix} x_{1} & x_{2} & \dots & x_{n} \end{bmatrix}^{\top} \\ & \mathbf{b} = \begin{bmatrix} b_{1} & b_{2} & \dots & b_{m} \end{bmatrix} \\ & \mathbf{c} = \begin{bmatrix} c_{1} & c_{2} & \dots & c_{n} \end{bmatrix} \\ & \mathbf{A} = \begin{bmatrix} a_{1,1} & a_{1,2} & \dots & a_{1,n} \\ a_{2,1} & a_{2,2} & \dots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \dots & a_{m,n} \\ \end{bmatrix} \end{array}