Here is the optimization problem:
For the function
$$ f(x_1,x_2;a_0,b_0)=\\\small\cases{\frac{1}{2}\left[x_1+x_2-x_1x_2+\left(\frac{-1+b_0(1-a_0)}{a_0}x_1+1\right)\left(\frac{-1+b_0(1-a_0)}{a_0}x_2+1\right)\right],\quad 0 \leq x_1\leq a_0,\, 0 \leq x_2\leq a_0\\ \frac{1}{2}\left[x_1+x_2-x_1x_2+\left(\frac{-1+b_0(1-a_0)}{a_0}x_1+1\right)b_0(-x_2+1)\right],\quad\quad\,\,\,\,\,\quad 0 \leq x_1\leq a_0,\, a_0 \leq x_2\leq 1\\ \frac{1}{2}\left[x_1+x_2-x_1x_2+b_0(-x_1+1)\left(\frac{-1+b_0(1-a_0)}{a_0}x_2+1\right)\right],\quad\quad\,\,\,\,\quad a_0 \leq x_1\leq 1,\, 0 \leq x_2\leq a_0\\ \frac{1}{2}\left[x_1+x_2-x_1x_2+b_0(-x_1+1)b_0(-x_2+1)\right],\quad\quad\quad\quad\quad\quad\quad a_0 \leq x_1\leq 1,\, a_0 \leq x_2\leq 1}$$
and
$$f(x_1=x_2;a_0,b_0)=\cases{\frac{1}{2}\left[2x_1-{x_1}^2+\left(1+\frac{x_1(1+b_0(1-a_0))}{a_0}\right)^2\right],\quad 0 \leq x_1 \leq a_0\\\frac{1}{2}\left[2x_1+b_0^2(1-x_1)^2-x_1^2\right],\quad\quad\quad\quad\,\quad a_0 \leq x_1 \leq 1}$$
We have:
$$\max_{a_0,b_0\in[0,1]}\left[\min_{{x_1=x_2\in[0,1]}}f(x_1,x_2;a_0,b_0) -\min_{{x_1,x_2\in[0,1]}}f(x_1,x_2;a_0,b_0)\right]$$
Is there a better way than brute force search? If yes, which method suits best? I mean to have less computational complexity by the word "best".
My Idea: I am thinking that the outer maximization could be done exhaustively but the inner minimizations could be solved somehow analytically. I dont know how to automize the process.
For example for given $a_0$ and $b_0$, one can take the derivative and make it equal to zero and insert back in the original equation and find the minimims. The problem is that I dont know if taking derivative finds me minimum or maximum? Any ideas?