(i) Consider the problem

$$\begin{aligned} \operatorname{minimize} & f(x) \\ \text { subject to } & h(x)=b, \quad x \in X, \end{aligned}$$

where $f: \mathbb{R}^{n} \rightarrow \mathbb{R}, h: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}, X \subseteq \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$. State and prove the Lagrangian sufficiency theorem.

In each of the following cases, where $n=2, m=1$ and $X=\{(x, y): x, y \geqslant 0\}$, determine whether the Lagrangian sufficiency theorem can be applied to solve the problem:

(ii) Consider the problem in $\mathbb{R}^{n}$

$$\begin{aligned} \operatorname{minimize} & \frac{1}{2} x^{T} Q x+c^{T} x \\ \text { subject to } & A x=b \end{aligned}$$

where $Q$ is a positive-definite symmetric $n \times n$ matrix, $A$ is an $m \times n$ matrix, $c \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$. Explain how to reduce this problem to the solution of simultaneous linear equations.

Consider now the problem

$$\begin{aligned} \operatorname{minimize} & \frac{1}{2} x^{T} Q x+c^{T} x \\ \text { subject to } & A x \geqslant b \end{aligned}$$

Describe the active set method for its solution.

Consider the problem

$$\begin{aligned} \operatorname{minimize} &(x-a)^{2}+(y-b)^{2}+x y \\ \text { subject to } & 0 \leqslant x \leqslant 1 \text { and } 0 \leqslant y \leqslant 1 \end{aligned}$$

where $a, b \in \mathbb{R}$. Draw a diagram partitioning the $(a, b)$-plane into regions according to which constraints are active at the minimum.

$$\begin{aligned} & \text { (a) } \quad f(x, y)=-x, \quad h(x, y)=x^{2}+y^{2}, \quad b=1 \text {; } \\ & \text { (b) } \quad f(x, y)=e^{-x y}, \quad h(x)=x, \quad b=0 \text {. } \end{aligned}$$