L13 - Linear Programming A general way for solving problem...

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Linear Programming A general way for solving problem What is a linear program (LP)? . 0 , , 8 2 4 3 11 2 4 5 3 2 subject to 3 4 5 maximize 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 + + + + + + + + x x x x x x x x x x x x x x x . 0 , , 8 2 4 3 - 11 2 4 - 5 3 2 - subject to 3 4 5 - minimize 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 - - - - - - - - - - - x x x x x x x x x x x x x x x or equivalently
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An Example We have two sources of protein, each  pound of peanut butter gives a  unit of protein, and each pound of  steak gives two units.  At least  four units are required in the diet.  Suppose a pound of peanut butter  costs $2 and a pound of steak  costs $3.  How much steak and  peanut should we buy such that we have the required units of  protein, and we pay the minimum cost. Suppose we buy x units of peanut butter, and y units of steak. 0 , where 4 2 subject to 3 2 minimize + + y x y x y x
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An Example We have two sources of protein, each  pound of peanut butter gives a  unit of protein, and each pound of  steak gives two units.  At least  four units are required in the diet.  Suppose a pound of peanut butter  costs $2 and a pound of steak  costs $3.  How much steak and  peanut should we buy such that we have the required units of  protein, and we pay the minimum cost. Suppose we buy x units of peanut butter, and y units of steak. 0 , where 4 2 subject to 3 2 minimize + + y x y x y x
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Many of the problems we have studied in this course can be formulated as linear programs. t 4 3 2 1 s 16 9 8 20 12 7 10 13 5 0 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( ) , ( subject to maximize 4 42 34 34 23 13 3 2 23 42 12 13 12 1 3 1 = + - = - + + = + - + = + - + ij t s t s ij ij s s f f f f f f f f f f f f f f f E j i c f f f This is the capacity constraints. To save space, I did not type them out. Not exactly the same; it should be ' ' here The maximum flow problem.
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It can be proved that the following programs are equivalent 0 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( ) , ( subject to maximize 4 42 34 34 23 13 3 2 23 42 12 13 12 1 3 1 = + - = - + + = + - + = + - + ij t s t s ij ij s s f f f f f f f f f f f f f f f E j i c f f f 0 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( ) , ( maximize 4 2 3 1 4 42 34 34 23 13 3 2 23 42 12 13 12 1 - + + - + - - + + + - + + - ij ts t t s s ts t s t s ij ij ts f f f f f f f f f f f f f f f f f f f f f E j i c f f Introduce a new variable to make thing symmetric Add two constraints for the source s and the sink t.
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Designing efficient algorithms for solving linear programs is very important. In fact, three mathematicians became very famous just because they have designed efficient algorithms for LP. simplex algorithm G.B. Dantzig 1950's Ellipsoid algorithm L.G. Khachian 1970's Interior point algorithm Karmarkar 1980's
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The simplex algorithm We will only describe the simplex algorithm. The other two involve too much mathematics.
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Just like the Ford-Fulkerson algorithm for maximum flow, the basic idea for the simplex algorithm is successive improvement .
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This note was uploaded on 07/05/2008 for the course CS CSIS0250 taught by Professor Dr.hing-fungting during the Summer '08 term at HKU.

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L13 - Linear Programming A general way for solving problem...

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