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Lecture 21 - Duality Theory II

Lecture 21 - Duality Theory II - Recap choosing the generic...

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Lecture 21 Duality Theory II UCSD Math 171A: Numerical Optimization Philip E. Gill http://ccom.ucsd.edu/~peg/math171 Monday, March 2nd, 2009 Recap: choosing the generic constraint format 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 n > m , i.e., G = Duality theory can turn a bad shape into a good shape UCSD Center for Computational Mathematics Slide 2/30, Monday, March 2nd, 2009 Recap: Choice of the primal problem 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 3/30, Monday, March 2nd, 2009 Primal problem 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 In standard form, the dual problem is minimize y R m - b T y subject to A T y = c , y 0 Basic idea: solve the dual and define the primal solution from the dual solution. UCSD Center for Computational Mathematics Slide 4/30, Monday, March 2nd, 2009
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The strong duality theorem implies that Final dual basic set = Final primal working set If B is the optimal basis for the dual, then A T B 4 = A T A & % A A = B T defines an optimal working-set matrix for the primal. UCSD Center for Computational Mathematics Slide 5/30, Monday, March 2nd, 2009 The dual problem from the previous slide is: minimize y R m - b T y subject to A T y = c , y 0 The optimal π -values for the dual satisfy “ B T π * = c B ”, where B ” = A T and c B ” = - b Substituting these values gives A π * = - b A ( - π * ) = b x * = - π * UCSD Center for Computational Mathematics Slide 6/30, Monday, March 2nd, 2009 The dual problem from the previous slide is: minimize y R m - b T y subject to A T y = c , y 0 The optimal reduced costs satisfy “ z ” = “ c - A T π * , where A ” = A T , c ” = - b and π * ” = - x * This gives the optimal reduced costs of the dual as: z ” = “ c - A T π * = - b - A ( - x * ) = Ax * - b 0 z * = r * , the optimal residual. the primal optimal residuals are the dual reduced costs. UCSD Center for Computational Mathematics Slide 7/30, Monday, March 2nd, 2009 Summary (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 Solution x * Multipliers λ * Residual r * Working set A Solution y * π -values π * Reduced costs z * Basis B The primal and dual solutions satisfy the following: x * = - π * λ * = y * ( λ * = y * B ) r * = z * A = B T UCSD Center for Computational Mathematics Slide 8/30, Monday, March 2nd, 2009
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Feasibility and boundedness The strong duality theorem states that if one of the primal or dual problems has a bounded
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