{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# hw1 - k x = exp-x T x is not 4 Plot the function f x = cos...

This preview shows page 1. Sign up to view the full content.

22M:270 Optimization Techniques Homework 1 — due Friday Jan 27 1. Find all the critical points of the following function f ( x , y ) = x 4 - x 2 y + 2 y 2 - 2 y . Determine if they are local minima, local maxima, or saddle points, by look- ing at the Hessian matrices at the critical points. Evaluate the function at the critical points. Can you determine if the function is coercive (that is, f ( x , y ) if p x 2 + y 2 )? Do you think that you have found the global minimum? Explain why. 2. Repeat Q1 for the following function f ( x , y ) = x 3 + 2 xy + y 2 . 3. Suppose that g : R n R is convex and f : R R is convex and increasing. Show that f g : R n R is convex. Use this to show that h ( x ) = exp ( x T x ) is convex; show as a counterexample that
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: k ( x ) = exp (-x T x ) is not. 4. Plot the function f ( x ) = cos ( 3 x )-. 5sin ( 5 x )+ . 2cos ( 10 x-π / 4 ) . Esti-mate how many critical points it has on the interval [ , 2 π ] . Now consider g : R 20 → R given by g ( x ) = f ( x 1 ) f ( x 2 ) ··· f ( x 20 ) . Show that ∇ g ( x ) = is equivalent to f ( x i ) = 0 for all i . From this, estimate how many critical points g has in the cube [ , 2 π ] 20 . Clearly there is a potential to have many (irrelevant) local minima. Can you suggest ways of handling problems like this? 1...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online