Calculate max/min of $x_1 x_2+y_1 y_2+z_1 z_2+w_1 w_2$

Question

What is a good way to calculate max/min of $$x_1 x_2+y_1 y_2+z_1 z_2+w_1 w_2$$ where $x_1+y_1+z_1+w_1=a$ and $x_2+y_2+z_2+w_2=b$ and $x, y, z, w, a, b \in \mathbb{N} \cup \{0 \}$, and please explain your answer (how your result comes out).

Answer 1

We can start with the inequality

\begin{equation} 0 \leq x_1x_2+y_1y_2+z_1z_2+w_1w_2 \leq (x_1+y_1+z_1+w_1)(x_2+y_2+z_2+w_2) = ab \end{equation}

Since you want a solution in $\mathbb{N}$ including $0$, you get equality only when $x_1 =a$ and $x_2=b$. The other variables will be zero. You get similar solutions by shifting the variables around.

To minimize the expression, you will have $x_1 = a$ and either $y_2$, $z_2$ or $w_2$ set to $b$ and all other variables set to zero.

Answer 2

I take it your natural numbers start at $1$. To maximize, let $x_1=a-3$, $x_2=b-3$, all other variables $1$. To minimize, $x_1=a-3$, $y_2=b-3$, all other variables $1$.