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Lecture 2 - Properties of Linear Constraints

Lecture 2 - Properties of Linear Constraints - Recap a...

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Lecture 2 Properties of Linear Constraints UCSD Math 171A: Numerical Optimization Philip E. Gill ( [email protected] ) Wednesday, January 7th, 2009 Recap: a simple linear program minimize x , y - 3 x - 4 y subject to 3 x + 2 y 200 x + 2 y 100 x 0 y 0 minimize/maximize = “optimize” UCSD Center for Computational Mathematics Slide 2/39, Wednesday, January 7th, 2009 Levels of equal profit 10 20 30 40 60 50 70 80 90 100 110 120 10 20 30 40 50 60 70 80 90 100 110 120 y x p = 0 p = 120 p = 240 p = 300 UCSD Center for Computational Mathematics Slide 3/39, Wednesday, January 7th, 2009 10 20 30 40 60 50 70 80 90 100 110 120 10 20 30 40 50 60 70 80 90 100 110 120 y x Optimal solution UCSD Center for Computational Mathematics Slide 4/39, Wednesday, January 7th, 2009
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Basic result of linear programming The optimal point ( x * , y * ) lies on the boundary point that is a “corner-point” of the feasible region. UCSD Center for Computational Mathematics Slide 5/39, Wednesday, January 7th, 2009 The “optimal” corner point lies at the intersection of the two lines 3 x + 2 y = 200 (cranberry juice constraint) x + 2 y = 100 (apple juice constraint) These are two equations in the two unknowns x * and y * . The solution is to mix: x * = 50 gallons of cranApple juice y * = 25 gallons of appleBerry juice With a profit of: 3 x * + 4 y * = 3 · 50 + 4 · 25 = 250/ c UCSD Center for Computational Mathematics Slide 6/39, Wednesday, January 7th, 2009 For a general linear program (LP) there may be thousands of variables. we need to write an LP in “generic form” Assume that there are n variables, x 1 , x 2 , . . . , x n and m constraints. UCSD Center for Computational Mathematics Slide 7/39, Wednesday, January 7th, 2009 minimize x 1 , x 2 ,..., x n c 1 x 1 + c 2 x 2 + · · · + c n x n subject to a 11 x 1 + a 12 x 2 + · · · + a 1 n x n b 1 a 21 x 1 + a 22 x 2 + · · · + a 2 n x n b 2 . . . . . . . . . a m 1 x 1 + a m 2 x 2 + · · · + a mn x n b m UCSD Center for Computational Mathematics Slide 8/39, Wednesday, January 7th, 2009
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Define x 4 = x 1 x 2 . . . x n and c 4 = c 1 c 2 . . . c n then x and c are vectors in R n , and c 1 x 1 + c 2 x 2 + · · · + c n x n = n X i =1 c i x i = c T x = x T c where c T x denotes the inner product of c with x . UCSD Center for Computational Mathematics Slide 9/39, Wednesday, January 7th, 2009 Similarly, if a i 4 = a i 1 a i 2 .
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