I am working on the following exercise:
Consider the following LP:
$$\min 24x_1-9x_2+8x_4 $$
such that \begin{align} 4x_1-9x_2+3x_3+4x_4 &= 2 \\ 3x_1-x_2-10x_3-x_4 &= 1 \\ x_1,x_2,x_3,x_4 &\ge 0 \end{align}
Draw the the corresponding column vectors $a_1, a_2, a_3, a_4, b$ as well as the dual feasible solution set graphically. Obtain a dual optimal solution from the sketch and use it to compute a primal optimal solution.
I have drawn the column vectors with geogebra. But I do not understand how I should now draw the dual feasible solution set and so on. Could you explain that to me?
To find the dual you turn the Min to a Max, flip the directions of the inequalities (a Min’s standard form has $\geq$’s), turn variables into constraints and turn constraints into variables. This is with an inequality leading to a non-negative variable and an equality leading to a free variable. This is why the column vectors are important; they are describing the constraints of the dual problem. With this in mind, the dual has the following form: $$\begin{align} \text{Max } 2y_1+y_2,\text{st:}&\\ 4y_1+3y_2\leq 24,&\\ -9y_1-1y_2\leq -9,&\\ 3y_1-10y_2\leq 0,&\\ 4y_1-1y_2\leq 8,&\\ y_1,y_2\ \text{free}. \end{align}$$ Since this linear programming problem only has two variables, this dual problem can have its feasible region can be plotted according to the above.
I hope this helps! Many thanks