lecture10 - IE 495 Lecture 10 Properties of the Recourse...

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IE 495 – Lecture 10 Properties of the Recourse Function Prof. Jeff Linderoth February 12, 2003 February 10, 2003 Stochastic Programming – Lecture 10 Slide 1
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Today’s Topic M E T H O D February 10, 2003 Stochastic Programming – Lecture 10 Slide 2
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Outline Small amount of review KKT/Optimality conditions for two-stage stochastic LP w/recourse The LShaped algorithm ƒ Example February 10, 2003 Stochastic Programming – Lecture 10 Slide 3
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Is That Your Final Answer What is the subgradient inequality? What are the KKT conditions? Q ( x, ω ) = min y ∈< p + { q T y : Wy = h ( ω ) - T ( ω ) x } . ƒ Name a vector s ∂Q ( x, ω ). Q ( x ) = E ω Q ( x, ω ) = s S p s Q ( x, ω ). ƒ Name a vector s ∈ Q ( x ). What is the Deterministic Equivalent of a stochastic program? February 10, 2003 Stochastic Programming – Lecture 10 Slide 4
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Our Favorite Problem min x ∈< n + : Ax = b ( c T x + E ω " min y ∈< p + { q T y : Wy = h ( ω ) - T ( ω ) x } #) min x ∈< n + : Ax = b ' c T x + E ω Q ( x, ω ) min x ∈< n + { c T x + Q ( x ) : Ax = b } February 10, 2003 Stochastic Programming – Lecture 10 Slide 5
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Highlights from Chapter 9 Q ( x, ω ) v ( h ( ω ) - T ( ω ) x ) is convex. Q ( x ) E ω Q ( x, ω ) is convex Q ( x ) is Lipschitz-continuous. If min y ∈< p + { q T y : Wy = h ( ω ) - T ( ω ) x } has unique dual solution λ * , then Q ( x, ω ) = - λ * T T If Q ( x ) = s S p s Q ( x, ω s ), then η = - X s S p s λ * T s T ( ω s ) Q ( x ) February 10, 2003 Stochastic Programming – Lecture 10 Slide 6
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Continuous Discussion
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  • Spring '08
  • Linderoth
  • Derivative, Trigraph, Convex function, Jensen's inequality, Convex analysis

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