Quiz1Review - Introduction to Mathematical Programming...

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Introduction to Mathematical Programming IE496 Quiz 1 Review Dr. Ted Ralphs
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IE496 Quiz 1 Review 1 Reading for The Quiz Material covered in detail in lecture. 1.1, 1.4, 2.1-2.6, 3.1-3.3, 3.5 Background material you should know. 1.2, 1.3, 1.5 Material you should read and be familiar with. 1.6, 2.7, 3.7 Optional material that may help your comprehension. 2.8, 3.6
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IE496 Quiz 1 Review 2 How to Study for The Quiz You should have done all the reading and understand the theorems and proofs in the sections we covered in detail. Review and understand the homework solutions. Review the lecture slides and make sure you can answer the questions that are posed. Go through the review slides and make sure you understand the flow of ideas and the connections among them. Ask questions if something is not clear.
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IE496 Quiz 1 Review 3 Math Programming Models A mathematical model consists of: Decision variables Constraints Objective Function Parameters and Data The general form of a math programming model is: min or max f ( x 1 ,...,x n ) s.t. g i ( x 1 ,...,x n ) = b i We might also want the values of the variables to belong to discrete set X .
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IE496 Quiz 1 Review 4 Linear Functions A linear function f : R n R is a weighted sum, written as f ( x 1 ,...,x n ) = n X i =1 c i x i for given coefficients c 1 ,...,c n . We can write x 1 ,...,x n and c 1 ,...,c n as vectors x,c R n to obtain: f ( x ) = c T x In this way, a linear function can be represented simply as a vector.
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IE496 Quiz 1 Review 5 Linear Programs A linear program is a math program that can be written in the following form: minimize c T x s . t . a T i x b i i M 1 a T i x b i i M 2 a T i x = b i i M 3 x j 0 j N 1 x j 0 j N 2 This in turn can be written equivalently as minimize c T x s . t . Ax b
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IE496 Quiz 1 Review 6 Standard Form To solve a linear program, we must put it in the following standard form : min c T x s.t. Ax = b x 0 Eliminate free variables : Replace x j with x + j - x - j , where x + j ,x - j 0 . Eliminate inequality constraints : Given an inequality a T i x b i , we can rewrite it as: a T i x - s i = b i s i 0
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IE496 Quiz 1 Review 7 A Little Linear Algebra Review Definition 1. A finite collection of vectors x 1 ,...,x k R n is linearly dependent if λ R k such that k i =1 λ i x i = 0 . Otherwise, the vectors are linearly independent . Let A be a square matrix. Then, the following statements are equivalent: The matrix A is invertible. The matrix A T is invertible. The determinant of A is nonzero. The rows of
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This note was uploaded on 02/29/2008 for the course IE 406 taught by Professor Ralphs during the Spring '08 term at Lehigh University .

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Quiz1Review - Introduction to Mathematical Programming...

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