Lecture 21 - Duality Theory II

Lecture 21- - Lecture 21 Duality Theory II UCSD Math 171A Numerical Optimization Philip E Gill http/ccom.ucsd.edu/~peg/math171 Monday March 2nd

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Unformatted text preview: 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 ≥ 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 ≥ In standard form, the dual problem is minimize y ∈ R m- b T y subject to A T y = c , y ≥ 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 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 ≥ 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 ≥ 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 ≥ ⇒ 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 ≥ 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...
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This note was uploaded on 10/23/2010 for the course MATH 171a taught by Professor Staff during the Winter '08 term at UCSD.

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Lecture 21- - Lecture 21 Duality Theory II UCSD Math 171A Numerical Optimization Philip E Gill http/ccom.ucsd.edu/~peg/math171 Monday March 2nd

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