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Lecture 22 - Taking the Dual

Lecture 22 - Taking the Dual - Recap duality Lecture 22...

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Lecture 22 Taking the Dual UCSD Math 171A: Numerical Optimization Philip E. Gill http://ccom.ucsd.edu/~peg/math171 Friday, March 6th, 2009 Recap: duality Primal in all-inequality form : (P): minimize x R n c T x subject to Ax b (D): maximize y R m b T y subject to A T y = c , y 0 Primal in standard form : (P): minimize x R n c T x subject to Ax = b , x 0 (D): maximize y R m b T y subject to A T y c UCSD Center for Computational Mathematics Slide 2/33, Friday, March 6th, 2009 Rules for primal-dual conversion Taking the dual ” involves three steps: 1 Convert the given LP to a generic all-inequality or standard form for the primal 2 “Take the dual”. Define the dual problem in terms of the generic quantities 3 Simplify UCSD Center for Computational Mathematics Slide 3/33, Friday, March 6th, 2009 Consider the primal in all-inequality form: minimize x R n c T x subject to Ax b A schematic representation of the problem data is " c T A b # The schematic form of the dual, in standard form , is then " - b T - A T - c # To see this we write down the dual: minimize y R m ( - b ) T y subject to ( - A ) T y = ( - c ) , y 0 UCSD Center for Computational Mathematics Slide 4/33, Friday, March 6th, 2009
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From the previous slide: minimize y R m ( - b ) T y subject to ( - A ) T y = ( - c ) , y 0 Now we simplify, to give minimize y R m - b T y subject to A T y = c , y 0 which is the same as maximize y R m b T y subject to A T y = c , y 0 UCSD Center for Computational Mathematics Slide 5/33, Friday, March 6th, 2009 Now consider the primal in standard form : minimize x R n c T x subject to Ax = b , x 0 A schematic representation of the problem data is " c T A b # The dual, in all-inequality form , has data " - b T - A T - c # which gives the dual problem: minimize y R m ( - b ) T y subject to ( - A ) T y ( - c ) UCSD Center for Computational Mathematics Slide 6/33, Friday, March 6th, 2009 From the previous slide: minimize y R m ( - b ) T y subject to ( - A ) T y ( - c ) Now we simplify, maximize y R m b T y subject to A T y c UCSD Center for Computational Mathematics Slide 7/33, Friday, March 6th, 2009 Example Next we form the dual of a linear program that is not already in one of the two generic primal forms. Consider the problem minimize w R n d T w subject to Gw f , w 0 UCSD Center for Computational Mathematics Slide 8/33, Friday, March 6th, 2009
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From the previous slide is: minimize w R n d T w subject to Gw f , w 0 Convert. We choose to write the primal in all-inequality form G I n w f 0 This problem has data: d T G f I n 0 UCSD Center for Computational Mathematics Slide 9/33, Friday, March 6th, 2009 Example Take the dual.
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