## Description

1. Minimize the function

f(x1, x2) = 100

x2 − x

2

1

2

+ (1 − x1)

2

using the following:

(a) 1

1 Random walk method with direction exploitation, using an initial guess of your /2

choice.

(b) Univariate method. 2

(c) 2

1 Conjugate directions (Powell’s) method. /2

Use an initial guess X1 =

−0.5

0.5

for the Univariate and Powell’s methods. Plot

the contour lines of the objective function and superimpose the directions along which

algorithm progresses for all the methods. Repeat the exercise using 2 additional initial

guess points of your choice for both the Univariate and Powell’s methods.

Note: You are encouraged to try out various initial guess points, and based on your

experience, present the ones which you find interesting. You are also encouraged to

comment on the differences (if you find any) in which the algorithm progresses based on

the choice of your initial guess points.

2. (a) 1 Compare the number of steps taken by the Univariate and the Powell’s methods in /2

question 1, for the choice of parameters you’ve taken (initial step length etc.).

(b) 1 Comment on the differences in which the Powell’s method progresses for the two /2

objective functions:

f(x1, x2) = (x1 + 2×2 − 7)2 + (2×1 + x2 − 5)2

and

f(x1, x2) = 100

x2 − x

2

1

2

+ (1 − x1)

2

3. Minimize the function f (x1, x2) = 8x 8

2

1 − 6x1x2 + 8x

2

2 − x1 + x2 using

(a) Steepest descent method

(b) Conjugate gradient method

(c) Newton’s method

(d) Marquardt method

For all the methods, assume the initial guess to be (100,0). For Marquardt method,

a = 104 where J˜ = J + aI (J and I are the Hessian and Identity matrices respectively).

Assume the other parameters appropriately. Perform a maximum of 4 iterations for

each method and compare them in terms of final accuracy. Mark the values of the

design variables on the contour plot of the objective function and tabulate the values at

each step for all the methods. Commment on your observation(s).

4. Minimize the function f(x, y) = x 5

2+y

2

subject to the inequality constraint x+2y ≥ 6 and

the equality constraint x − y = 3 using interior and exterior penalty function methods.