Mathematical optimization with thresholded optimization function
I guess $f_2(x)$ is some sort of oracle that tells you if $\Phi_2(x)$ is greater than $t$?  If the set $\{\mathbf{x} : \Phi_2(x) \le t\}$ is convex, then I think projected gradient algorithm might be helpful. Could you provide more information on $\Phi_2(x)$? The question is kinda confusing with the information provided.