Gomory Mixed-Integer Set

$X^{GOM}=\left\{\left(y_0,y,x\right)\in {\mathbb{Z}}^1\times {\mathbb{Z}}_{+}^p\times {\mathbb{R}}_{+}^q:y_0+{\sum }_j{a}_j{y}_j+{\sum }_k{g}_k{x}_k=b\right\}$

mit $b\notin {\mathbb{Z}}^1$.

Umformung in Continous Integer Knapsack Set: $\left\{\left(y_0,y,s\right)\in {\mathbb{Z}}^1\times {\mathbb{Z}}_{+}^p\times {\mathbb{R}}_{+}^1:y_0+{\sum }_j{a}_j{y}_j\le b+s\right\}$

mit MIR Ungleichung:

$y_0+{\sum }_j\left(\lfloor a_j\rfloor +\frac{{\left(f_j-f_b\right)}^{+}}{1-f_b}\right)y_j\le \lfloor b\rfloor -{\sum }_{k:g_k<0}\frac{g_k}{1-f_b}{x}_k.$ → Gomory Mixed-Integer Ungleichung