I am new to optimization and I to solve the following optimization problem:
$$ \min_{g_i(x) \leq 0, h_i(x) = 0} \max_{ \{p_1, \dots,p_n \} \in \Delta} \sum p_i - p_i x_i $$
where the constraints induce convex sets. I thought about writing this equivalently to
$$ \min y $$ $$ y \leq 1 - x_i$$ $$ g_i(x) \leq 0 $$ $$ h_i(x) = 0 $$
and use KKT to solve this. But I am not sure how to proceed from here. Any tips?