ISyE 6664 Stochastic Optimization
Fall 2007
Assignment 3
Issued: September 20, 2007
Due: September 27, 2007
All problems in this assignment can be done in groups of 2.
Problem 1
We want to calculate the following expected value (d-dimensional integral) of fd(Ļ), where
Ļis uniformly distributed on the hypercube [0,10]d:
Ed[fd(Ļ)] = ī[0,10]d
d
ī
i=1 ī10
āĻi
+Ļisin(Ļ2
i)ī1
10ddĻ
Note that this integral decomposes into a separate integral for each dimension, and is easy
to calculate exactly. Calculate the exact value of Ed[fd(Ļ)]. Now pretend that the integral
does not decompose. The objective is to empirically compare the eļ¬ciency of diļ¬erent
computational methods for approximating Ed[fd(Ļ)]. Use a multidimensional numerical
integration method of your choice, such as the trapezoid method or Simpsonās method.
Note: Do not simply calculate the one-dimensional integrals with a numerical integration
method and then use the results. Although that would be correct for this particular integral
because of its simple form, the point of the exercise is to compare the accuracy and eļ¬ciency
of diļ¬erent methods for computing multidimensional integrals. Also use a random sampling
method (Monte Carlo integration). Compute Ed[fd(Ļ)] with these methods, for d=1,2,...,
(see how large you can make dwith less than 2 hours of computing time) and for varying
number nof points Ļjwhere the function fd(Ļj) is evaluated. For each of these combinations
of method, d,andn, compute the root square error (which you can do exactly, but which is
random for the sampling method). For the sampling method, you may want to obtain a more
accurate estimate of the root mean square error by replicating the experiment a number of
times. Make plots of the root mean square error versus n, and compare the eļ¬ciency of the
methods for diļ¬erent d.
