Homework 3

Q1. Quadratic functions

• Beck, Exercise 2.19

Q2. Homogeneity of the directional derivative

(a) Prove that the directional derivative (when it exists) of a function $f:\Re^n\to\Re$ is positively homogeneous of degree one, that is, show that for all $\alpha\ge0$,

Use this fact to argue that the directions $d$ in which the directional derivative is evaluated can always be normalized, so that $\|d\|_2=1$.

(b) Which unit-norm direction minimizes the directional derivative, i.e., which direction provides the largest instantaneous decrease in $f$ at the point $x$?

Q3. Nonlinear least squares (Beck, Exercise 4.6)

(a) Derive the gradient $\nabla f(x)$. (Note that this is a slightly different objective than used in the formulation discribed in lecture. Yes, you could use AD to compute this, but here we're asking you to derive it analytically.)

(b) Part (ii) of this problem.