I wrote this question on Maths.stackexchange but perhaps this community suits better (?)

I need to ask you for this question, which is a rather general one, in order to understand how to behave when studying maxima and minima with constraints, in many variables. The specific question is the following: suppose I have some $f(x, y, z)$ (in this case I'm specifically asking for three variables) subject to two constraint: $$ \begin{cases} g(x, y, z) \leq 0 \\\\ h(x, y, z) = 0\end{cases}$$

  • Question 1: since I have mixed constraints (an equality and an inequality constraint), should I still study internal points (when referring to $g$), hence $\nabla f(x, y, z) = (0, 0, 0)$ or not?

  • Question 2: I then use Langrange multipliers in the usual way, by building the Lagrangian

$$L(x, y, z, \lambda, \mu) = f(x, y, z) - \lambda g(x, y, z) - \mu h(x, y, z)$$

(then I know how to proceed).

Say I will be able to reduce the problem to a two dimensiona one, and I cannot go further than this. Then I study $\nabla f(x, y) = (0, 0)$ and I hopefully find some points. What now? Should I use them to find the third point $z$ by using the constraints, or should I study $f(x, y)$ with the Hessian matrix?

Thank you!


Just to make things more concrete, I thought to write here an example for what I meant. Say $$f(x, y, z) = xyz$$

Subject to $$\begin{cases} x^2+y^2+z^2 \leq 1 \\\\ x+y+z = 1 \end{cases}$$

So a sphere and a plane, which intersect to form a cicumference.

Internal points

$$\nabla f(x, y, z) = (0, 0, 0)$$ returns points of the form

$$(x, 0, 0) \qquad \qquad (0, y, 0) \qquad \qquad (0, 0, z)$$

Hence on the axis. And here the question one: should I consider them, or should I interpret the two constraints as the circumference and stop?

In all those points, $f(\cdot)= 0$.

Going on:

$$L(x, y, z, \lambda, \mu) = xyz - \lambda(x^2+y^2+z^2-1) - \mu(x+y+z-1)$$

And the derived equations read

$$ \begin{cases} yz = 2\lambda x + \mu \\ xz = 2\lambda y + \mu \\ xy = 2\lambda z + \mu \\ \text{the two constraints} \end{cases} $$

I was able to write down:

$$\mu = zy - 2\lambda x$$ hence from arranging the second with this one:

$$\lambda = \frac{-z}{2}$$

And then subtituting all in the third:

$$xy = -z^2 + zy + zx$$

Using the second constraint for $z \to z = 1-x-y$ I conclude with

$$f(x, y) = 2x^2+ 2y^2-3x-3y+xy +1$$

So at this point: $\nabla f(x, y) = (0, 0)$ gives the points

$$x = 0 \qquad \qquad y = 0$$

$$x = \frac{3}{5} \qquad \qquad y = \frac{3}{5}$$

and fron this, question two: should I find $z$ and then simply evaluate $f$ on this points, or should I go on with $f(x, y)$ with eh Hessian?

  • Bonus question: am I reasoning right or wrong? Am I missing something?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.