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Lecture1

# Lecture1 - C&O 355 Mathematical Programming Fall 2010...

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Mathematical Programming Fall 2010 Lecture 1 N. Harvey

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What is Optimization? Basic Idea: From a certain set of objects, choose the best one. Example: Getting married “objects” = { all possible spouses } “best one”… this is very hard to make precise
What is Mathematical Optimization? Additional Idea: Describe objects by vectors in R n Example: Each possible spouse described by a vector in R 4 Best could mean “highest IQ” or “fewest spouses”, etc. Age 32 30 67 93 Prev Spouses 1 1 0 9 IQ 152 200 109 122 Eye Color 1 2 1 3 (Using the Encoding: 1 = Brown, 2 = Blue, 3 = Green)

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What is Mathematical Optimization? Additional Idea: Describe objects by vectors in R n Example: We can formalize the “highest IQ spouse” problem as: Age 32 30 67 93 Prev Spouses 1 1 0 9 IQ 152 200 109 122 Eye Color 1 2 1 3
A Continuous Problem Often there are infinitely many objects from which we want to find the best one. Example: Allocating assets x 1 = Dollar value of my Walmart shares x 2 = Dollar value of my US treasury bonds Here f : R 2 ! R is some function that measures “risk”

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Definition: A mathematical program is an optimization problem that looks like this: where x 2 R n , f : R n ! R , g i : R n ! R , b i 2 R objective function
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Lecture1 - C&O 355 Mathematical Programming Fall 2010...

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