## Description

1. The variable y represents the yield in a chemical process. There are n process variables x1, x2, . . . , xn

(such as temperature, flow rate, etc) which influence the yield. Data was collected to observe

the yield y for various values of the process vector x

t = (x

t

1

, xt

2

, . . . , xt

n

). It is believed that y can

be reasonably approximated by a convex quadratic objective function. Formulate the problem

of finding the best convex quadratic approximation Q(x) for y using the available data as a

nonlinear program, and discuss the important features of your formulation.

To test your model, ensure that you write GAMS statements to generate random inputs that

are consistent with the above hypothesis. Also, write statements to print out the results of the

model to show how the solution relates to your random inputs. [Hint: you may want to use the

fact that if Q = RT R then Q is guaranteed to be symmetric positive semidefinite.]

2. Consider a truss consisting of two steel tubes pinned together at one end and fixed at two pivot

points at the other end. The span – the distance between the two pivots is fixed at 2s. The

problem is to choose the height of the truss and the thickness and average diameter of the steel

tubes so that the truss will support a load of 2W and so that the total weight of the truss will

be minimized. Denote the average tube diameter by d, the tube thickness t and the truss height

by h. What is the weight of the truss? (assume density of steel is 1).

The following constraints must be observed:

• Because of space limitations the height of the truss must not exceed b1.

• The ratio of the diameter of the tube to the thickness of the tube must not exceed b2.

• The compression stress is limited by yield stress:

W(s

2 + h

2

)

1/2 ≤ b3dth

• The tube must not buckle:

W(s

2 + h

2

)

3/2 ≤ b4dh(d

2 + t

2

)

(a) Formulate this problem as a standard nonlinear program and solve using GAMS. Experiment with various starting points or solvers to ensure you find a good solution.

scalar W / 15 /, b1 /2/, b2 /3/, b3 /0.5/, b4 /1/;

scalar s /1/;

3. Suppose you have a number of spheres of different radii with their centers given. Find the

radius and center of the smallest sphere that encloses all these spheres. Ensure you use a

SOCP formulation for this.

Use the following GAMS code to generate a data set to test your formulation:

Problem 0 Page 1

CS635 Problem Set #10

$if not set n $set n 2000

set i /1*%n%/;

set j /1*3/;

* generate random reproducable data

option seed = 101;

parameter r(i); r(i) = uniform(0,0.2);

parameter c(i,j); c(i,j) = uniform(0,1);

4. Find the area of the largest n-gon (polyhedron with n vertices) that can be inscribed within the

unit circle. Note that the solution is know to be unsymmetric for n odd. Find the (approximately)

best areas for n = 15, 25, 35.

Problem 0 Page 2