(i) Consider the problem

$$\begin{aligned} \text { minimise } & f(x) \\ \text { subject to } & g(x)=b, x \in X, \end{aligned}$$

where $f: \mathbb{R}^{n} \longrightarrow \mathbb{R}, g: \mathbb{R}^{n} \longrightarrow \mathbb{R}^{m}, X \subseteq \mathbb{R}^{n}, x \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$. State the Lagrange Sufficiency Theorem for problem $(*)$. What is meant by saying that this problem is strong Lagrangian? How is this related to the Lagrange Sufficiency Theorem? Define a supporting hyperplane and state a condition guaranteeing that problem $(*)$ is strong Lagrangian.

(ii) Define the terms flow, divergence, circulation, potential and differential for a network with nodes $N$ and $\operatorname{arcs} A$.

State the feasible differential problem for a network with span intervals $D(j)=$ $\left[d^{-}(j), d^{+}(j)\right], j \in A .$

State, without proof, the Feasible Differential Theorem.

[You must carefully define all quantities used in your statements.]

Show that the network below does not support a feasible differential.

Part II 2004