Description
1. Problem 3.1.1.a) and 3.1.1.b) . For both problems, you need to study and
use the Second Order Sufficiency Condition in Proposition 3.2.1 to verify
that your solution is indeed a local minimum. (15%)
2. Industrial design. A cylindrical can is to hold 4 cubic inches of orange
juice. The cost per square inch of constructing the metal top and bottom is
twice the cost per square inch of constructing the cardboard side. What are
the dimensions of the least expensive can? (15%)
3. Duality. Read Section 3.4 and study Example 3.4.2. Prove that the
following two linear programs are dual to each other
Min c’x , subject to A’x ≥ b
Max b’μ, subject to Aμ = c, μ ≥ 0 (15%)
4. Problem 4.2.1 (a) (b) and (d) (15%)
Hint: The augmented Lagrangian function with quadratic penalty is
described in pages 398-404.
5. Mathematical modeling for data mining. (40%)
Linear regression is one of the fundamental models for data mining. The
model describes a linear relationship between a number of numerical
attributes x= (x1, x2, …, xn) and a predicted variable y in the form of
y = 𝒂’x+b,
where a ∈ R
n
and b ∈ R are parameters to be determined by training.
The
training process takes a set of K training examples
(X, Y) = {(x
1
, y1
), (x
2
, y2
), …, (x
K, yK
)},
where each x
i∈ R
n
is a vector of attributes.
The parameters a and b are
determined by minimizing the mean squared error (MSE):
MSE = ∑
K
i=1
[y
i
– (a’x
i+b)]2
Build a linear regression for the following program effort data. Each
training sample consists of an index of social setting, an index of family
planning effort, and the percentage change in the crude birth rate (CBR)
between 1965 and 1975, for 20 countries in Latin America.
Here, we want to
predict change (y) using setting (x1) and effort (x2). Therefore, we have that
n = 2 and K =20.
setting(x1) effort(x2) change(y)
Bolivia 46 0 1
Brazil 74 0 10
Chile 89 16 29
Colombia 77 16 25
CostaRica 84 21 29
Cuba 89 15 40
DominicanRep 68 14 21
Ecuador 70 6 0
ElSalvador 60 13 13
Guatemala 55 9 4
Haiti 35 3 0
Honduras 51 7 7
Jamaica 87 23 21
Mexico 83 4 9
Nicaragua 68 0 7
Panama 84 19 22
Paraguay 74 3 6
Peru 73 0 2
TrinidadTobago 84 15 29
Venezuela 91 7 11
Write an AMPL model for the optimization problem, and submit it to NEOS
to obtain the optimal parameters a and b in the linear regression model. You
need to choose a suitable solver in NEOS.
You cannot use any other existing
software for linear regression. Submit the following:
1) The AMPL model file (and data file, if any)
2) A print-out of the solution from your NEOS solver.
3) A table listing the model error y
i
– (a’x
i+b) for all the 20 countries.
4) Discuss the insights you gained from this analysis, such as: How does
each attribute influence the change? Which attribute seems to have stronger
correlation with the change? Does the linear regression model seem accurate
to you?
