Math/CS 661: Optimization --- Fall 2008

To contact the instructor: j.rhodes@uaf.edu

Course syllabus: M661syl.pdf

Homework Assignments:

The dates below are when problems were assigned. Unless otherwise noted, problems are always due the following Monday.

Problems are from Nocedal & Wright, unless otherwise stated.

• 9/5 (due 9/15) Use the definitions of convexity to 1) Show the subset of R^2 with y>=x^2 is convex, and the subset with y<=x^2 is not convex. 2) Show f(x)=x^2 is convex on R and g(x,y)=x^2+y^2 is convex on R^2, but h(x,y)=x^2-y^2 is not convex on R^2, nor is -h(x,y). 3) Show that for any vector a in R^n, and vector x of n variables, the linear function l(x)=a•x is convex on R^n.
• 9/8 problems 2.1, 2.2, 2.3, 2.7, 2.8
• 9/10 no new problems
• 9/12 no new problems
• 9/15 2.9, 3.2, 3.4
• 9/17 No class
• 9/19 no new problems
• 9/22 problems 2.13, 2.14, 2.15, 2.16
• 9/24 problem 3.1 (also explain what you see happening for each algorithm and starting point, by referring to the plot or level curves of the Rosenbrock function), 3.7
• 9/26 no new problems
• 9/29 problems 3.3, 3.5, 3.6, 5.2, 5.3 and 1) Develop a Gram-schmidt-like process that for any given symmetric, positive definite A and vectors v_1,v_2,...v_k will produce a set of vectors with the same span that are conjugate with respect to A. 2) Demonstrate your process on the matrix A=[1 1 0; 1 2 1; 0 1 2] and vectors (1,0,0), (0,1,0), (0,0,1).
• 10/1 problems 5.1, 5.4, 5.5, 5.6
• 10/3 No class
• 10/6 No new problems
• 10/8 No new problems
• 10/10 (due 10/20) 5.6, and 1) Construct some large sparse positive definite matries A with eigenvalues clustered (or not clustered) in several ways. Then produce experimental runs of the CG algorithm to solve Ax=b for some b, showing its performance at each step. You might present the runs with figures like 5.4 and 5.5 in the text, or in some other way that makes the same point.