how do you solve the following optimization problem to find the global solution?
$~~~~~\underset{y}{min} ~ \underset{x}{max} f(x,y)$
subject to
$~~~~~g(x)<0$
with knowing that both g(x) and f(x,y) are convex in the domain of x and y. the g only depends on x. What is the best approach to perform this optimization problem? of course one possible way is to iterate between x and y,i.e. pick y and then solve the maximization over x and then use the found x to solve the minimization over y, and repeat. But this method does not guarantee global minimum.
For the example, take the following function which are convex in both x and y.
$f(x,y)=x^Ty + x^Tx.y^Ty~~~$ (bilinear terms)
$g(x)=x^Tx-1$