Linear Optimization

A linear programming problem consists of a linear function to be maximized (or minimized) subject to linear equality and inequality constraints. Any linear program (LP) can be put by well-known transformations into standard form (11.1)$$\begin{array}{l} (LP)\begin{array}{*{20}{c}} z&{{c^T}x}&{ = \max }\\ {}&{{A_x}}&{ = b}\\ {}&x&{ \ge 0} \end{array}\\ \mathop \Leftrightarrow \limits_{\max \{ {c^T}x|x \in X\} ,X: = \{ x \in {R^n}|Ax = b,x \ge 0\} ,} \end{array}$$ where A is a real m x n matrix, $$ b \in {\mathbb{R}^m},\,c\, \in \,{\mathbb{R}^n}$$. The input data of (11.1) are given by the triple $$P = (A,b,c)\, \in \,{\mathbb{R}^{m \cdot n + m + n}}$$.