辅导案例-MAST30013-Assignment 2
UNIVERSITY OF MELBOURNE SCHOOL OF MATHEMATICS AND STATISTICS MAST30013 Techniques in Operations Research Semester 1, 2020 Assignment 2 Due: 4 pm, Thursday, 7 May - Solution must be typeset in LaTex. - Please submit your solution online by the due date. - Show all necessary working. 1. Consider the function f : R4 ! R: f(x) = x41 + x 4 2 + x 4 3 x31x3 12x1x23 + x1x2x3 3x32 4x1x2 + x24 + 9x2 3x4 + 4. You are required to do a computational study comparing the below three methods for finding the local and global minima of f . i Steepest descent method; ii Newton’s method; iii BFGS Quasi-Newton method. (a) Compute the gradient and hessian of f . (b) Create a set of instances which consists of 1000 randomly generated initial points for the algorithms. Test the algorithms on the instance set and compare their average performance in terms of solutions found and computational time. Use the following parameters: • tolerance ✏1 = 10 2 for the three methods, • tolerance ✏2 = 10 5 for the Golden section search, • step size T = 10, • initial points with coordinate values range xi 2 ( 10, 10), i = 1, 2, 3, 4. i. You should report the average performance of your algorithms in tables. f value Minimiser No. of times Ave iterations per search Ave time per search (sec)