class_8_linear_programming(2).pdf

# class_8_linear_programming(2).pdf - Linear programming...

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Linear programming Linear programming Linear problems Standard form and Slack form The simplex algorithm A few facts regarding the simplex algorithm (without proofs) Linear programming August 12, 2017

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Linear programming Linear programming Linear problems Standard form and Slack form The simplex algorithm A few facts regarding the simplex algorithm (without proofs) Overview 1 Linear programming Linear problems Standard form and Slack form The simplex algorithm A few facts regarding the simplex algorithm (without proofs)
Linear programming Linear programming Linear problems Standard form and Slack form The simplex algorithm A few facts regarding the simplex algorithm (without proofs) Linear programming Many problems can be naturally described as maximization/minimization of a linear function under linear constrains. For example: Alice and Bob are flying to Paris for a vacation. They want to take with them as many books as possible , however they can’t carry more then 12 pounds of books. Alice will take x 1 books and Bob will take x 2 books. Alice’s books weigh 3 pounds each, and Bob’s books weigh 1 pound each. After hours of discussions, they decided that given the length of their vacation, and how fast they read, they should make sure that: x 2 2 - 3 4 x 1 2 3 x 2 + 10 3

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Linear programming Linear programming Linear problems Standard form and Slack form The simplex algorithm A few facts regarding the simplex algorithm (without proofs) Linear programming Many problems can be naturally described as maximization/minimization of a linear function under linear constrains. For example: Alice and Bob are flying to Paris for a vacation. They want to take with them as many books as possible , however they can’t carry more then 12 pounds of books. Alice will take x 1 books and Bob will take x 2 books. Alice’s books weigh 3 pounds each, and Bob’s books weigh 1 pound each. After hours of discussions, they decided that given the length of their vacation, and how fast they read, they should make sure that: x 2 2 - 3 4 x 1 2 3 x 2 + 10 3
Linear programming Linear programming Linear problems Standard form and Slack form The simplex algorithm A few facts regarding the simplex algorithm (without proofs) Linear programming We can express the problem as: Maximize: x 1 + x 2 subject to the linear constrains: 3 x 1 + x 2 12 3 x 1 - 2 x 2 10 4 x 1 - 2 x 2 ≥ - 3 x 1 , x 2 0

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Linear programming Linear programming Linear problems Standard form and Slack form The simplex algorithm A few facts regarding the simplex algorithm (without proofs) Linear programming Another example: We can express the problem of finding maximal flow in a flow network as follows: Maximize: X v V f ( s , v ) - X v V f ( v , s ) subject to: f ( u , v ) c ( u , v ) for any ( u , v ) V × V X v V f ( u , v ) = X v V f ( v , u ) for any u V - { s , t } f ( u , v ) 0 for any ( u , v ) V × V (Here we assume all edges exist and the capacities of edges that don’t actually exist are 0.)
Linear programming Linear programming Linear problems Standard form and Slack form

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