## Description

Optimization: Homework 2

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 ∈ Rn and b ∈ R are parameters to be determined by training. The training process takes a set of K training examples

(X, Y) = {(x1, y1), (x2, y2), …, (xK, yK)},

where each xi ∈ Rn is a vector of attributes. The parameters a and b are determined by minimizing the mean squared error (MSE):

MSE =

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 yi – (a’xi+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?

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