Q1: Consider the function f: R2 → R given by

f(x1, x₂)

(a) Find all stationary points of f.

mum/maximum or a saddle point.

x²₁x₁x₂ + x² + x₁-x2.

Classify each point, i.e., determine if it is a local mini-

(b) Starting at x = (0,0), find the local minimum of f using Newton's method (algorithm

below).

Stop when ||Vf(x, ak)|| <0.01.

For the optimal step size use Matlab or similar to find the minimum of g(a) = f(x*+adk).

You may also use Matlab or similar to work with the gradient and Hessian of f.

[15 marks]

Newton's Method

1: Start from an initial point º, set k=0.

2: If (Vf(x) < ) then exit.

If V2f(a) is positive definite then

dk-[V2 f(x)]-¹Vf(x)

else

dk = -f(x).

3: Calculate ak = minazo f(x + adk).

4: Set æk+1=k+ad and k=k+1 then return to step 2.