Optimum Design of Mechanical Elements and Systems
Problem 1: For the following functions:
find the Taylor series about (1,1) up to quadratic term.
Then, using the Taylor series,
predict the function value at (1.1, 0.8). Compare against exact value.
Problem 2: For the generic quadratic function:
classify the stationary point (minimum, maximum, saddle,
'other') based on the
coefficients. Using this general expression, classify the stationary points of
Problem 3: For the following quadratic functions:
find the stationary point(s) (using MATLAB's ezmesh,
ezcontour, fminunc, …), and
Problem 4: Consider the function:
Find the minimum using fminunc. Now find the minimum
within the region
0 < u < 0.5 and 0 < v < 0.1 using a naïve random search using K = 1,00,0000
(maximum) random searches. Plot the error (distance from exact to result from
random search) for K =100, K =1000, K = 10000, K = 100,000 and K =
1,00,0000. There is no need to submit your Matlab code; just submit the plot.
Problem 5: Consider the function f (x,y) = 3x^2−xy
+ 4y^2−x −y . Find the
minimum by performing line-search in two directions, first in x then in y, starting at
(0,0). (You need to do this problem by hand, but you are welcome to confirm your
answers with the line-search code.)
Problem 6: Consider the function f (x,y) = x^2 −xy
+ 4y^2 + x −3 . Suppose a
direction d = (1,1) is given. Find the corresponding conjugate direction. Using this
pair show that you can find the minimum from the origin in two steps.
Problem 7: Write a Matlab function to find the
minima of an arbitrary N-D function
via the Powell's method (you may use the line-search method provided, but at your
own risk!). Your code should be of the form:
function [xMin,fMin,iter] = ...
where the function parameters are as before. Test your
code against a variety of
quadratic and non-quadratic functions. Email only the Powell code (and not the test scripts).