# Nonlinear Optimization Math 524, Fall 1996

### Professor Robert Sharpleysharpley@math.sc.eduDepartment of MathematicsUniversity of South Carolina

• 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)
• 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