A convex quadratic program

Question

Let

\begin{align*} f &:= 4-x-y\\ g &:= x+y+5 \end{align*}

We want to maximize $fg$ subject to $f \ge 0$ and $g \ge 0$.

I tried to solve it using Lagrange multipliers but could not.

Answer 1

Lagrange multiplier is not really needed,

Let $p=x+y$, you want to $$ \max (4-p)(p+5)$$

subject to

$$4-p \geq 0 \iff p\leq 4$$

and

$$p+5 \geq 0 \iff p\geq -5$$

We can notice that the constraints are indeed redundant.

Can you solve $$\max (4-p)(p+5)?$$

Notice that this is just a quadratic problem.

After you solve for $p$. Choose $x=x_0$, you can solve for $y=p-x_0$.