We define $Q_{>0} = (\mathbb{R}_+^*)^{I \times J}$ and $\mathscr{L} : Q_{>0} \times (\mathbb{R}^I \times \mathbb{R}^J) \rightarrow \mathbb{R}$ defined by $$\mathscr{L}(\boldsymbol{q}, (f, g)) = J_\epsilon(\boldsymbol{q}) + \sum_{i \in I} f_i \left(\alpha_i - \sum_{j \in J} q_{ij}\right) + \sum_{j \in J} g_j \left(\beta_j - \sum_{i \in I} q_{ij}\right).$$ (a) Verify that $Q_{>0}$ is an open convex set of $\mathbb{R}^{I \times J}$. (b) Show that there exists $(f(\epsilon), g(\epsilon)) \in \mathbb{R}^I \times \mathbb{R}^J$ such that $\mathscr{L}(q(\epsilon), (f(\epsilon), g(\epsilon)))$ is a saddle point of $\mathscr{L}$. (Hint: One may identify $\mathbb{R}^{I \times J}$ with $\mathbb{R}^n$ and $\mathbb{R}^I \times \mathbb{R}^J$ with $\mathbb{R}^m$, for $n$ the cardinality of $I \times J$ and $m$ the sum of the cardinalities of $I$ and $J$, then use question 3 of part I.)
We define $Q_{>0} = (\mathbb{R}_+^*)^{I \times J}$ and $\mathscr{L} : Q_{>0} \times (\mathbb{R}^I \times \mathbb{R}^J) \rightarrow \mathbb{R}$ defined by
$$\mathscr{L}(\boldsymbol{q}, (f, g)) = J_\epsilon(\boldsymbol{q}) + \sum_{i \in I} f_i \left(\alpha_i - \sum_{j \in J} q_{ij}\right) + \sum_{j \in J} g_j \left(\beta_j - \sum_{i \in I} q_{ij}\right).$$
(a) Verify that $Q_{>0}$ is an open convex set of $\mathbb{R}^{I \times J}$.\\
(b) Show that there exists $(f(\epsilon), g(\epsilon)) \in \mathbb{R}^I \times \mathbb{R}^J$ such that $\mathscr{L}(q(\epsilon), (f(\epsilon), g(\epsilon)))$ is a saddle point of $\mathscr{L}$. (Hint: One may identify $\mathbb{R}^{I \times J}$ with $\mathbb{R}^n$ and $\mathbb{R}^I \times \mathbb{R}^J$ with $\mathbb{R}^m$, for $n$ the cardinality of $I \times J$ and $m$ the sum of the cardinalities of $I$ and $J$, then use question 3 of part I.)