Multivariate Optimization with some prior knowledge

Question

I have some question about multivariate optimization.

I want to solve this kind of optimization problem :

$minimize_{x, y} f(x,y)$ and what I have is that $f(x,y)$ has global minimum through some unknown interval $y \in [a,b]$ and of course corresponding argument minimum $x$ depends on y. I want to find $b$ which is the maximum value of $y$ where the global minimum is attained.

Now I just minimize $f$ with respect to $x, y$ both and maximizing y. But this method is a heuristic and not good enough in my opinion. Do you have any idea?

Answer 1

One way would be to do this in two steps:

1. Minimize $z=f(x,y)$. Record the optimal objective function value $z^*$.
2. Maximize $y$ subject to $f(x,y)=z^*$.