Unit 4

Question 1

Note

Answer 1

Gradient is a vector valued function with domain and codomain in R^n

Question 2

Necessary condition for there to be a minimum at x*?

Answer 2

The FOC is satisfied (=0n)

Question 3

Pulling it all together: solution method? (2 steps and 2 hessian outcomes)

Answer 3

1) find any critical points
2) obtain hessian matrix H(x)

If hessian is PD for ALL x, objective function is combs and any critical point yields a unique global minimum

If hessian is PD at x* (but not all x) then there is a strict local minimum at x*

Question 4

Note:

Answer 4

Revise first year matrix work (eg. Cramer’s rule) for simplicity

Question 5

See notes

Answer 5

Subscript notation or partial derivatives

Question 6

What do Taylor’s polynomials do?

Answer 6

They use Taylor’s theorem to give information about a function f in the REGION of a chosen point (ie. x*), however it sacrifices ‘global’ info about f

Question 7

With Taylor polynomials, what should you do if possible?

Answer 7

Put your answer in a form so the bit in the brackets are all the same factors (see class 3 question 1)

Question 8

Quick way to identify if symmetric 2x2 matrix is positive definite?

Answer 8

Given A= a  b
                 b c
If a>0 and ac>b^2 then matrix A is positive definite (know from class 2)