Optimization with Boundary Solutions

Question

I am trying to maximize this objective function $$f(x)=-x^2$$ subject to $1 \le x\le 3 $

This is the Langrangian I wrote:

$$\mathcal L (x, \mu, \lambda) := x^2 + \lambda (3-x)+\mu (x-1)$$

F.O.C + other conditions

1) $2x=\mu-\lambda$

2) $\lambda (3-x)=0$

3) $\mu (x-1)=0$

4) $\lambda \geq0, \mu \ge0$

What am I stuggling with assuming that I dont know which constraints bind, how do I solve knowing potentially the answer is on the boundary? i.e, w/o converting this problem to a minimization or adding slack terms, how do I solve for the answer with the 4 equations I have? Thank you.

Answer 1

If the only permitted way of solving problem is using Lagrange multiplier, you are going to consider all possibilities for Lagrange multipliers , for example from relation 2. you either have $[\lambda > 0 , ~ x=3]$ or $\lambda = 0$

First case gives you $x=3.$

Second case gives you $ 2x = \mu $ using 3 you get $x=0 ,$ or $x=1$. ($x=0$ is not feasible)

So all candidate points for optimality are $$x= 1, ~ 3$$

Last step: Check them to find maximum point!

Answer 2

Guide:

There are only $2$ boundaries, you can try out

Case $1$: $x=1$.

Case $2$: $x=3$.

Case $3$: optimal $x$ is in the interior.

Answer 3

HINT: the searched Maximum is $f(1)$ since your function is monotonously decreasing in the given interval