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handout23 - Recap Choice of the generic form Math 171A...

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Math 171A: Linear Programming Lecture 23 Linear Programming Duality Philip E. Gill c 2011 http://ccom.ucsd.edu/~peg/math171a Friday, March 4th, 2011 Recap: Choice of the generic form minimize w R n d T w subject to Gw f , w 0 Convert to all-inequality form if m > n , i.e., G = Convert to standard form if m < n , i.e., G = Duality theory can turn a bad shape into a good shape UCSD Center for Computational Mathematics Slide 2/35, Friday, March 4th, 2011 Problem conversion involves defining a primal linear program and converting it into another dual linear program First, we define the primal problem as an LP in all-inequality form: minimize x R n c T x subject to Ax b where A has m rows. UCSD Center for Computational Mathematics Slide 3/35, Friday, March 4th, 2011 Primal problem in all-inequality form
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First, we define the primal problem as an LP in all-inequality form: minimize x R n c T x subject to Ax b where A has m rows. Assume for the moment that the problem is feasible with a bounded optimal value of c T x * . UCSD Center for Computational Mathematics Slide 5/35, Friday, March 4th, 2011 The optimality conditions are: c = A T λ * , λ * 0; λ * · ( Ax * - b ) = 0 Suppose the Lagrange multipliers are viewed as particular values of dual variables λ i . The optimality conditions A T λ * = c and λ * 0 become the values of dual constraints A T λ = c and λ 0 evaluated at λ = λ * . UCSD Center for Computational Mathematics Slide 6/35, Friday, March 4th, 2011 Any λ such that A T λ = c and λ 0 is said to be dual feasible . Any x such that Ax b is said to be primal feasible . UCSD Center for Computational Mathematics Slide 7/35, Friday, March 4th, 2011 If x is primal-feasible (i.e., Ax b ) and λ is dual-feasible (i.e., A T λ = c and λ 0) then λ T ( Ax - b ) = λ T |{z} 0 ( Ax - b | {z } 0 ) 0 Some rearrangement gives λ T ( Ax - b ) = λ T Ax - λ T b = c T x - b T λ 0 . UCSD Center for Computational Mathematics Slide 8/35, Friday, March 4th, 2011
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It follows that for a given x such that Ax b , c T x b T λ for all λ such that A T λ = c , and λ 0. UCSD Center for Computational Mathematics Slide 9/35, Friday, March 4th, 2011 The smallest value of c T x is c T x * . It follows that c T x * b T λ for all λ such that A T λ = c , and λ 0 The largest value of b T λ is defined by the linear program maximize λ R m b T λ subject to A T λ = c , λ 0 . This is called the dual linear program . UCSD Center for Computational Mathematics Slide 10/35, Friday, March 4th, 2011 The dual linear program is maximize λ R m b T λ subject to A T λ = c , λ 0 . The constraints of this dual problem are in standard form The dual constraint matrix is the transpose of the primal constraint matrix. UCSD Center for Computational Mathematics Slide 11/35, Friday, March 4th, 2011 For the moment we assume that the dual problem is feasible.
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