Formulating an optimisation problem into a mixed-integer problem

Question

I'm not sure if I understand this question and was wondering if anyone could provide any insight to an answer. The only thing I can think of adding is a constraint: "x2 = integer", so I'm clearly missing something. Any help would be appreciated :)

Answer 1

To have a Linear Mixed Integer Programming, replace the second constraint by $$ 4x_1+4(y_1+3y_2+4y_3+10y_4)\ge 10\\ y_1+y_2+y_3+y_4=1\\ y_1,y_2,y_3,y_4 \in \{0,1\} $$

With $y_1+y_2+y_3+y_4=1$, you force exactly one term of the sum $y_1+3y_2+4y_3+10y_4$ to be active. The active term is precisely $x_2$.

Answer 2

A special case of integer variables are binary variables. The second constraint can be reformulated as

$y_1\cdot x_2+\frac{1}{3}\cdot y_2\cdot x_2+\frac{1}{4}\cdot y_3\cdot x_2+\frac{1}{10}\cdot y_4\cdot x_2=1$

$y_1, y_2, y_3, y_4\in \{0,1 \}$

Answer 3

This is also a correct formulation. $$ 4x_1+4(y_1+3y_2+4y_3+10y_4)\ge 10\\ y_1+y_2+y_3+y_4\le 1\\ y_1,y_2,y_3,y_4 \in \{0,1\}\\ x_1 \ge 0 $$

$y_1+y_2+y_3+y_4 = 1 $ ( avoiding $x_1 \ge 0 $ ) also works as mentioned previously due to redundant constraints.