Task $1$

Given the following mixed integer LP:

min,$2x_1-5x_2$

$u.d.N.-8x_1+7x_{2}20$

$,x_1+x_{2}8$

$,x_1,x_{2}0$

$,x_1,x_2$inZ

Solve the MILP using the branch$-$and$-$bound method. Illustrate the evolution of the solutions via tree structure.

Task $2$

Write a commented MATLAB program that solves an MILP in general for using the branch$-$and$-$bound method. Use linprog to solve the relaxed LP$.$ In addition, your program should satisfy the following specifications:

After each iteration:

Output the number of all nodes to be examined.

output the current vector $x$

output the current value $c^{T}x$

After convergence:

$\frac{1}{2}$

Output of the required iterations.

Output of the optimal vector $x^{**}$

Output the optimal value $c^T{x}^{**}.$

Solve with your program:

The MILP from task $1$