CS-544 Optimization in Computer Vision

1. Show that the point (1, 1) is the only local minimizer of this function, and that the Hessian matrix at that point is positive definite
2. Implement the line search algorithm in the notes (pages from Nocedal and Wright, Algorithm 3.5), using whatever programming language and tools you favor (I like Matlab for this).
3. Find the minimum, using gradient descent and line search, starting at (1.2, 1.2).
4. Find the minimum, using gradient descent and line search, starting at (-1.2, 1).
5. Find the minimum, using Newton's method and line search, starting at (1.2, 1.2).
6. Find the minimum, using Newton's method and line search, starting at (-1.2, 1).
7. Use your skill and judgement to compare approaches and start points (one should be more difficult than the other), and explain why you observed what you observe. To make this argument, you might: plot the step lengths; plot the steps; consider the time taken; plot the function; etc. I'm looking for a reasoned argument, with evidence, rather than a guess.

You should do this exercise individually. Submit PDF's to me by Feb 21. Email them to me at daf@uiuc.edu, with "CS544" in the header