I think I am not understanding some key steps to apply the LM method when we have several constrains. For instance, if we had to find the global extrema of a function $f(x,y,z)$ in a set M defined by the constrains $g_1(x,y,z)$ and $g_2(x,y,z)$, and an additional condition, say $ g_3 =y\le0$, I would follow these steps:
- Check the interior of M for critical points that are in the interior (through the gradient of f)
- Apply Lagrange to $g_1$
- Apply Lagrange to $g_2$
- Apply Lagrange to $g_3$
- Apply Lagrange to the intersections $$g_{1} \cap g_{2} \cap g_{3}$$ $$g_{1} \cap g_{2}$$$$g_{1} \cap g_{3}$$$$g_{2} \cap g_{3}$$
- Select the maximum and minimum by computing the values of f at the points.
However, sometimes my professor just does Steps 1, 5A; while other times he goes through every step.
I am really confused.
All help is deeply appreciated.
If your set $M$ is defined by the constraints $$ g_1 (x,y,z) \leq0 \qquad g_2 (x,y,z) \leq0 $$
Then we can find a higher dimensional function $$ g(x,y,z) = (g_1(x,y,z),g_2(x,y,z)) $$
and then $M$ is defined by the constraint $$ g(x,y,z) \leq0, $$ so we have just one (although higher-dimensional) function.
That is why it is very often only considered, where the set is only defined by one inequality.