Mathematical Optimization — Assignment 2 - IFOR

Institute for Operations Research
ETH Zurich HG G21-23
Kathrin Ballerstein kathrin.ballerstein@ifor.math.ethz.ch
Martin Ballerstein martin.ballerstein@ifor.math.ethz.ch
Dr. Alberto Del Pia alberto.del.pia@ifor.math.ethz.ch
Spring Semester 2012
Mathematical Optimization — Assignment 2
http://www.ifor.math.ethz.ch/teaching/Courses/Spring_2012/Mathematical_Optimization
Assignment 1: Linear Interpolation (6 points)
A physicist investigates the relation between two quantities x and y. She knows that y is a function
of x, but she is unsure about the exact relationship. Therefore, she has conducted an experiment and
obtained a series of pairs (x i ,y i ), i = 1,...,m. The physicist conjectures that y is a linear function of
x and decides to examine her conjecture with the help of linear programming. Her task is to determine
the “best” line y = ax+b such that the sum of the deviations of the predicted values on the line and
the observed values in the experiment is minimal.
(a) Find a linear program which is able to compute the desired values for a and b and which does not
use more than m+2 variables.
(b) Find a linear program which is able to compute the desired values for a and b and which does
not use more than m non-trivial equations or inequalities. An equation (resp. inequality) is called
trivial if it is of the form s = 0 (resp. s ≥ 0 or s ≤ 0) for some variable s.
The experiment provides the following result:
x 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
y 1.0 0.9 0.7 1.5 2.0 2.4 3.2 2.0 2.7 3.5
(c) Compute a and b with the help of an LP-solver and present the data and the “best” line graphically.
Hints
In (a) and (b), you may start from the following non-linear optimization problem:
m∑
min |ax i +b−y i | s.t. a,b ∈ IR.
i=1
For (c) you can use the website http://www.ifor.math.ethz.ch/cplex/index.en.html.
Assignment 2: Removal planning (4 points)
A mathematician intends to move to another flat and needs some help to decide whether this task
can be performed. He has n items of weight a 1 ,...,a n . To convey the items they are put into boxes.
Available are m boxes of maximal input weight b 1 ,...,b m . After putting the items into the boxes
they must be placed on a van with maximum permitted load Q. Formulate an integer linear program
which allows a decision whether the removal is possible or not.
Assignment 3: Separating hyperplanes (6 points)
(a) Given are m points (x i ,a i ), i = 1,...,m, such that x i ∈ IR n and a i ∈ {0,1}. The aim is to decide
whether it is possible to separate the points x i with a hyperplane c T x = 1 such that all points
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with a i = 0 satisfy c T x i ≤ 1 and all points with a i = 1 satisfy c T x i > 1. Formulate the decision
problem as a linear optimization problem.
(b) Given are m points (x i ,y i ), i = 1,...,m, such that x i ∈ IR n and y i ∈ IR. The aim is to find
vectors b 1 , b 2 , and c such that the hyperplane c T x = 1 partitions the m points into two parts and
the following expression is minimized:
∑ ∑
|y i −b T 1x i |+ |y i −b T 2x i |.
{i:c T x i≤1}
{i:c T x i>1}
Formulate a mixed-integer linear optimization problem to determine the vectors b 1 , b 2 , and c.
(Due date: Friday, March 9th, 2012, before the exercise class)
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