lecture9

lecture9 - IE 495 Lecture 9 Properties of the Recourse...

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IE 495 – Lecture 9 Properties of the Recourse Function Prof. Jeff Linderoth February 10, 2003 February 5, 2003 Stochastic Programming – Lecture 9 Slide 1

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Outline a Two-stage stochastic LP Convexity Continuity Differentiability Optimality Conditions ? L-Shaped Method! February 5, 2003 Stochastic Programming – Lecture 9 Slide 2
A Bit of Review minimize c T x + E ω £ q T y / subject to Ax = b T ( ω ) x + Wy ( ω ) = h ( ω ) ω Ω x < n + y ( ω ) < p + Q ( x, ω ) = min y ∈< p + { q T y : Wy = h ( ω ) - T ( ω ) x } February 5, 2003 Stochastic Programming – Lecture 9 Slide 3

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All the Same 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 v ( 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 5, 2003 Stochastic Programming – Lecture 9 Slide 4
Proofs If LP duality holds... v ( z ) = min y ∈< p + { q T y : Wy = z } = max t ∈< m { z T t : W T t q } Let Λ = { λ 1 , λ 2 , . . . , λ | Λ | } be the set of extreme points of { t ∈ < m | W T t q } . ƒ Each of those extreme points λ k is potentially an optimal solution to the LP. ƒ In fact, we are sure that there is no optimal solution better than one that occurs at an extreme point, so we can write... v ( z ) = max k =1 ,..., | Λ | { z T λ k } , z ∈ < m . February 5, 2003 Stochastic Programming – Lecture 9 Slide 5

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What’s All This? αv ( z 1 ) + (1 - α ) v ( z 2 ) = max k =1 , 2 ,... | Λ | z T 1 λ k + (1 - α ) max k =1 , 2 ,... | Λ | z T 2 λ k αz T 1 λ * k + (1 - α ) z T 2 λ * k = ( αz 1 + (1 - α ) z 2 ) T λ * k max k =1 , 2 ,... | Λ | [( αz 1 + (1 - α ) z 2 ) T λ k ] = v (( αz 1 + (1 - α ) z 2 )) Quite Enough Done ???? ? What did I just prove? February 5, 2003 Stochastic Programming – Lecture 9 Slide 6
I’m an Idiot! The above proof is hopelessly wrong Take z 1 , z 2 dom( v ) v (( αz 1 + (1 - α ) z 2 )) = max k =1 ,..., | Λ | { ( αz 1 + (1 - α ) z 2 ) T λ k } = ( αz 1 + (1 - α ) z 2 ) T λ k * = ( αz T 1 λ * k + (1 - α ) z T 2 λ * k α max k =1 ,..., | Λ | z T 1 λ k + (1 - α ) max k =1 ,..., | Λ | z T 2 λ k = αv ( z 1 ) + (1 - α ) v ( z 2 ) Quite Enough Done February 5, 2003 Stochastic Programming – Lecture 9 Slide 7

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What if LP duality doesn’t hold. We Make It Hold!
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• Spring '08
• Linderoth
• lim, Convex function, stochastic programming, Programming – Lecture

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