# EXPSPACE proof and its implications

I'm dealing with the min-max regret 0-1 Integer Linear Programming problem (MMR-ILP, for short), which is formulated as below.

$$\begin{equation} \label{eq:nip_obj} \min_{x \in \Phi} \sum_{i = 1}^n u_i x_i - \theta \end{equation}$$

$$\begin{equation} \label{eq:nip_constraints} \theta \leqslant \sum_{i = 1}^n \left(l_i + (u_i - l_i)x_i\right)y_i, \quad \forall~y \in \Phi \end{equation}$$

where $$\Phi$$ is the set of all possible solutions for the problem, $$x,y \in \Phi$$, $$\Theta \in \mathbb{R}_{>0}$$, and $$|\Phi|$$ is exponential.

I strongly believe that MMR-ILP is EXPSPACE-Complete (or, at least, EXPSPACE) since $$\Phi$$ is an exponential set (and not given by the problem instance/benchmark).

I'm not a theoretical scientist, sadly. I'm studying the polynomial hierarchy and the time and space hierarchies, but I did not find any good literature regarding this topic.

I want to show that this problem does not belong to NP. Thus, my questions are as follows.

1. Which technique I can use to show that it's in EXPSPACE?
2. If I show that MMR-ILP is in EXPSPACE, thus it implies that MMR-ILP does not belong to NP?
3. If I show that MMR-ILP is in EXPSPACE, what I can affirm regarding its time complexity (as far as I read, it's strongly believed that EXPTIME $$\subseteq$$ EXPSPACE)?