Students may work in groups of up to three people. You may consult only your textbooks, your notes, online information about AMPL, and me.

Due: Friday, December 10. (20 points)
In DDD’s original model, they chose b and w to

(1)

for the m = 5 factors and the set of I = {1,…, 40} compounds. In this part of the project, we look at alternative objectives.

If you prefer to use them, my model and run file are available on the course website. For each question in this part of the project, take C = 1, ϵ = 0.05, and use all 40 samples to find w and b.

Along with the solutions to the questions, hand in your new model files, highlighting the changes you made.

1. Points outside the tube: DDD wants to minimize the number of points outside the tube, so they use the objective:

   (2)

   where

   (3)

   for each i I. Model this problem as an integer program and solve it. (You can assume |y_i - b - ∑ _j=1^mw _jx_ij| ≤ 1 for each i I.)

2. Components of w:
   One aim of the term C ∑ _j=1^m|w _j| in the objective (1) is to try to sparsify w, that is, to encourage more zero components in w. As an alternative, DDD wants to explicitly limit the number of nonzero components of w to be no larger than 3, and use the objective function

   (4)

   Model this problem as an integer program and solve it. (You can assume |w_j| ≤ 5 for j = 1,…, 5.)

3. Both aspects:
   DDD wants to combine both modeling aspects above, so they want to use the objective

   (5)

   with κ defined in (3), and they want to impose the constraint that no more than 3 components of w be nonzero. Model this problem as an integer program and solve it. (You can assume |y_i - b - ∑ _j=1^mw _jx_ij| ≤ 1 for each i I and |w_j| ≤ 5 for j = 1,…, 5.)

AMPL hints:

• A variable can be declared binary. For example:
  var make {SET} binary;

• cplex is available for Windows or Mac OSX or Red Hat Linux: see
  http://www.rpi.edu/~mitchj/ILOG11.htm

  It is also available on RCS. Instructions for using it on RCS are available from

  http://www.rpi.edu/~mitchj/cplex11.html

  From a unix machine, use ssh rcs-sun.rpi.edu or ssh rcs-ibm.rpi.edu to access RCS. From a Windows machine, you can use SecureCRT; you need to connect to rcs-sun or rcs-ibm to ensure that cplex and ampl are available.

• Alternatively, if you don't have cplex running, you can submit your model to the NEOS server at

  http://www-neos.mcs.anl.gov/neos/

  I would suggest using either the solver cbc or the solver MINTO. Note that you will need to use a run file if you want to get output that is nicely formatted and easy to interpret.