I'm facing this problem:
$$ \large \min_{x \in \mathbb{R}_+^3} \max \left\{ { \sum_{i=1}^3 x_i^2-2 x_1 x_3 \over \left(\sum_{i=1}^3 x_i \right)^2} , { \sum_{i=1}^3 x_i^2 + 2 (x_1 x_3 - x_1x_2+x_2x_3) \over \left(\sum_{i=1}^3 x_i \right)^2} \right\} $$
I don't know how to deal with inner $\max$ and choose one of two!
I'm trying to use $max(A,B) \geq \frac12(A+B)$! Do you have any idea?
Thanks
A standard trick when dealing with max-min or min-max problem is to introduce a new decision variable, call it $z$, which models the inner optimization. Hence, your objective function becomes $$ \min_{z \in \mathbb{R}, \bf{x} \in \mathbb{R}^3_+} z $$ with the following constraints $$\begin{aligned} z &\geq \dfrac{\sum_{i=1}^3 x_i^2-2 x_1 x_3}{\left(\sum_{i=1}^3 x_i \right)^2} \\ z &\geq \dfrac{\sum_{i=1}^3 x_i^2 + 2 (x_1 x_3 - x_1x_2+x_2x_3)}{\left(\sum_{i=1}^3 x_i \right)^2} \\ \sum_{i=1}^3 x_i &= 1 \end{aligned}$$ The first two inequalities model the $\max$ expression, i.e. they state that whatever $z$ comes out to be, it has to be larger than either of the two terms. The last one captures the homogeneity, as it has been pointed out in the comments. From here, you can apply the KKT conditions.