Lecture10

# Lecture10 - Advanced Mathematical Programming IE417 Lecture...

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Unformatted text preview: Advanced Mathematical Programming IE417 Lecture 10 Dr. Ted Ralphs IE417 Lecture 10 1 Reading for This Lecture • Primary Reading – Chapter 6, Sections 2-3 IE417 Lecture 10 2 Saddle Point Optimality IE417 Lecture 10 3 Lagrangian Saddle Points • Recall the Lagrangian function Φ( x,μ,v ) = f ( x ) + μ T g ( x ) + v T h ( x ) • A point ( x * ,μ * ,v * ) with x * ∈ X,μ * ≥ is a saddle point for Φ( x,μ,v ) if Φ( x * ,μ,v ) ≤ Φ( x * ,μ * ,v * ) ≤ Φ( x,μ * ,v * ) , ∀ x ∈ X, ( μ,v ) ,μ ≥ . IE417 Lecture 10 4 Saddle Point Optimality • A point ( x * ,μ * ,v * ) with x * ∈ X,μ * ≥ is a saddle point for Φ( x,μ,v ) if and only if – Φ( x * ,μ * ,v * ) = min { Φ( x,μ * ,v * ) : x ∈ X } – g ( x * ) ≤ , h ( x * ) = 0 , and – μ * T g ( x * ) = 0 . • Furthermore, ( x * ,μ * ,v * ) is a saddle point if and only if x * and ( μ * ,v * ) are the optimal solutions to P and D with no duality gap, i.e., f ( x * ) = Θ( μ * ,v * ) . IE417 Lecture 10...
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Lecture10 - Advanced Mathematical Programming IE417 Lecture...

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