grandes-ecoles 2023 Q13

grandes-ecoles · France · x-ens-maths__psi Stationary points and optimisation Existence or properties of extrema via abstract/theoretical argument

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) Show that for all $(f, g) \in \mathbb{R}^I \times \mathbb{R}^J$, the minimum of $\boldsymbol{q} \mapsto \mathscr{L}(\boldsymbol{q}, (f, g))$ on $Q_{>0}$ is attained at $q(f,g)_{ij} = e^{(f_i + g_j - C_{ij})/\epsilon} p_{ij}$.
(b) Calculate the value of $G(f, g) = \mathscr{L}(q(f,g), (f,g))$.
(c) Verify that $G$ is concave on $\mathbb{R}^I \times \mathbb{R}^J$.

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) Show that for all $(f, g) \in \mathbb{R}^I \times \mathbb{R}^J$, the minimum of $\boldsymbol{q} \mapsto \mathscr{L}(\boldsymbol{q}, (f, g))$ on $Q_{>0}$ is attained at $q(f,g)_{ij} = e^{(f_i + g_j - C_{ij})/\epsilon} p_{ij}$.\\
(b) Calculate the value of $G(f, g) = \mathscr{L}(q(f,g), (f,g))$.\\
(c) Verify that $G$ is concave on $\mathbb{R}^I \times \mathbb{R}^J$.

Paper Questions

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