Nonlinear Optimization
Math 524, Fall 1996
Professor
Robert Sharpley
sharpley@math.sc.edu
Department of Mathematics
University of South Carolina
Table of Contents
Math524 Info and Syllabus -
(PostScript file)
Note: ---> Date for Test 3 changed to Monday, Nov. 25 <---
Dichotomous Search Example -
(PostScript file)
Golden Section Search Example -
(PostScript file)
Newton's Quadradic Search Method -
(PostScript file)
Geometric Programming Example -
(PostScript file)
Homework Assignments
Calculus Review Problems - p. 31: 1 (a,b,d)
Golden Section Search Problem - due Friday, 9/6/96
For the function f(x):= x^2 + 4 * cos(x), a=0, b=2:
(a) Perform 4 iterations by hand (calculator) using Golden Section Search.
(b) Check your work using the Matlab algorithms.
Newton Problem - due Wednesday, 9/11/96
For the function f(x):= x^2 + 4 * cos(x) and the starting values a=0, 1., 3, compute the Newton iterates using the matlab algorithm provided.
After running this experiment and analyzing the results, explain the behavior of the results.
Multivariate Problems (due Monday, 9/16/96)
page 32 of Text, Problems 4 (a)-(b) , 10 (b)
For each of these expressions, regard them as functions of the variables x_1, x_2, ... and determine all critical points.
At each critical point, determine the Hessian matrix.
Write the given expression in Taylor form as was given in class for the general form.
Multivariate Problems 2 - due Friday, 9/24/96
Homework problem from class: For the given matrix, provide the (a) LDL' factorization, (b) eigenvalues, (c) determinants of the principal minors to verify the equivalence theorem for positive definite matrices.
Pages 32-33: #7 (a)(c), #8 (a)(c), #9, #10 (b), [Extra Credit - #17]
Multivariate Problems 3 - due Friday, 10/4/96
Pages 32-33: #10 (b), #12 (a)(b)(c)(d)(f), #13
Multivariate Convexity - due Wednesday, 10/16/96
Page 77: #2 (all)
Page 78: #8 (all)
Geometric Programming I (see the handout above) - due Friday, 11/1/96
Page 77: #16 (a),(b)
Page 78: #18
Geometric Programming II - due Wednesday, 11/6/96
Page 81: #26 (Set up but do not solve)
Multivariate Newton Method (due Friday, 11/8/96) You may use the matlab script added below to assist in solving this problem.
Page 129: #4 (but compute to 12 decimal place accuracy)
Multivariate Plots and Newton's method. II (Due Monday, 11/11/96)
Same as previous problem, but apply to Rosenbrock's function: f(x,y) = 10 (x^2-y)^2 + (x-1)^2, -2 <= x,y <= 2.
Method of Steepest Descent - due Wednesday, 11/20/96
Using the function f(x,y):= 10 x^2 +xy +y^2, compute the iterates x^(1) and x^(2) with x^(0) = (1,0).
Miscellaneous Materials
Matlab
Matlab Primer -
(Postscript file)
Dichotomous Search algorithm -
(Matlab m file)
Golden Section Search algorithm -
(Matlab m file)
Newton iteration algorithm (1-D version)-
(Matlab m file)
Simple 2D plot script -
(Matlab m file)
Function for use with plotter -
(Matlab m file)
Multivariate plotter-contourer, with Newton algorithm to approximate critical points, and function examples -
(Matlab m file)
If you have any questions, please send e-mail to
sharpley@math.sc.edu
Last modified: 24 Sept. 1996