Max-min of a nonconvex quadratic problem
I assume that your "unit simplex" is the set of all nonnegative points whose coordinates sum to 1, $\{x: x_i\ge 0, \sum x_i = 0\}.$

Note that the max occurs when $x=y$ as $x_iy_i\le \left(\frac{x_i+y_i}2\right)^2$.  This makes your problem easy to solve, as at the optimal $x$ either $x_i^2-b_i$ is the solution, or $x_i=0$ and the solution is less than $-b_i$, so a binary search on the sorted $b_i$ should get your answer quickly.