Lagrange multipliers with bounds on variables

Question

Can we still use Lagrange multipliers if our variables are bounded? For example, we want to maximize $x^2 + y^2 + y^2$ subject to $x^4 + y^4 + z^4 = 1.$ This is very simple to approach by Lagrange Multipliers, but how can we use the method if we add bounds to the variables, let's say for example $x,y,z \in [0,2]$ ?

Answer 1

If you set your variables between 0 and $a$, one can simply adjusts the constrain $f(x)$ to $f(a-x)$

So, in this case, the problem is reformulated as

Maximise $x^2 + y^2 + z^2$ subject to $(2-x)^4 + (2-y)^4 + (2-z)^4 = 1$

which should be okay to solve. If the lower bound is not 0, then the problem becomes more complicated, though a solution is to introduce slack variables.

Answer 2

This is not really an answer but just a trick to bypass the problem.

This is small trick I was given almost fifty years ago when I faced a similar problem : if you want to impose $a \leq x\leq b$, define $$x=a+\frac{b-a}{1+e^{-X}}$$ It can make the problem more difficult to solve but, at least, you get rid of the problem with bounds.