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lecture20 - Duality Karush-Kuhn-Tucker and Lagrangian...

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April 25, 2006 IE447 – 1 Duality, Karush-Kuhn-Tucker and Lagrangian methods Aur´elie Thiele Lehigh University April 25, 2006 IE 417 Nonlinear Programming

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The Dual Problem Review on Duality Dual Problem Pros and Cons Favorable Structures April 25, 2006 IE447 – 2 s Consider the primal problem: min f ( x ) s.t. x X, g j ( x ) 0 , j = 1 , . . . , r, assuming it has a finite optimum.
The Dual Problem Review on Duality Dual Problem Pros and Cons Favorable Structures April 25, 2006 IE447 – 2 s Consider the primal problem: min f ( x ) s.t. x X, g j ( x ) 0 , j = 1 , . . . , r, assuming it has a finite optimum. s The dual problem is defined as: max q ( μ ) s.t. μ 0 , with dual function: q ( μ ) = inf x X { f ( x ) + μ g ( x ) b ² L Lagrangian }

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Pros and Cons for Solving the Dual Review on Duality Pros and Cons Pros Cons Favorable Structures April 25, 2006 IE447 – 3
Pros for Solving the Dual Review on Duality Pros and Cons Pros Cons Favorable Structures April 25, 2006 IE447 – 4 1. The dual is concave. By contrast, the primal may not be convex.

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Pros for Solving the Dual Review on Duality Pros and Cons Pros Cons Favorable Structures April 25, 2006 IE447 – 4 1. The dual is concave. By contrast, the primal may not be convex. 2. The dual may have smaller dimension, and has simpler constraints.
Pros for Solving the Dual Review on Duality Pros and Cons Pros Cons Favorable Structures April 25, 2006 IE447 – 4 1. The dual is concave. By contrast, the primal may not be convex. 2. The dual may have smaller dimension, and has simpler constraints. 3. If there is no duality gap and the dual is solved exactly for a Lagrange multiplier μ , all optimal primal solutions can be obtained by minimizing the Lagrangian L ( x, μ ) over x X .

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Pros for Solving the Dual Review on Duality Pros and Cons Pros Cons Favorable Structures April 25, 2006 IE447 – 4 1. The dual is concave. By contrast, the primal may not be convex. 2. The dual may have smaller dimension, and has simpler constraints. 3. If there is no duality gap and the dual is solved exactly for a Lagrange multiplier μ , all optimal primal solutions can be obtained by minimizing the Lagrangian L ( x, μ ) over x X . 4. This can also be useful to gain insights into the optimal solution. (More on this later.)
Pros for Solving the Dual Review on Duality Pros and Cons Pros Cons Favorable Structures April 25, 2006 IE447 – 4 1. The dual is concave. By contrast, the primal may not be convex. 2. The dual may have smaller dimension, and has simpler constraints. 3. If there is no duality gap and the dual is solved exactly for a Lagrange multiplier μ , all optimal primal solutions can be obtained by minimizing the Lagrangian L ( x, μ ) over x X .

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lecture20 - Duality Karush-Kuhn-Tucker and Lagrangian...

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