Optimization of Discrete Models

The basic model is:

$$\text{Optimize }{f}_j(\mathbf{X})\text{for }j\in J$$ $$\text{subject to }{g}_i(\mathbf{X})\{\begin{array}{l}\ge b_i\\ =b_i\\ \le b_i\end{array}\text{for all }i\in I$$

In the formula, to optimize means to maximize or minimize. The subscript $j$ represents to the number of functions that are needed to optimize. $J$ is a finite set, including integer subsctipts, $j$. $\mathbf{X}$: Decision Variables.