(i) Let $P$ be the problem

$$\text { minimize } f(x) \quad \text { subject to } h(x)=b, \quad x \in X \text {. }$$

Explain carefully what it means for the problem $P$ to be Strong Lagrangian.

Outline the main steps in a proof that a quadratic programming problem

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

where $Q$ is a symmetric positive semi-definite matrix, is Strong Lagrangian.

[You should carefully state the results you need, but should not prove them.]

(ii) Consider the quadratic programming problem:

$$\begin{aligned} \operatorname{minimize} & x_{1}^{2}+2 x_{1} x_{2}+2 x_{2}^{2}+x_{1}-4 x_{2} \\ \text { subject to } 3 x_{1}+2 x_{2} \leqslant 6, \quad x_{1}+x_{2} \geqslant 1 . \end{aligned}$$

State necessary and sufficient conditions for $\left(x_{1}, x_{2}\right)$ to be optimal, and use the activeset algorithm (explaining your steps briefly) to solve the problem starting with initial condition $(2,0)$. Demonstrate that the solution you have found is optimal by showing that it satisfies the necessary and sufficient conditions stated previously.