lecture15 - IE 495 Lecture 15 Bounds in Stochastic...

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IE 495 – Lecture 15 Bounds in Stochastic Programming Prof. Jeff Linderoth March 5, 2003 March 5, 2003 Stochastic Programming – Lecture 15 Slide 1
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Outline Review ƒ Jensen's Lower Bound ƒ What the @ ˆ !&* ˆ !*! went wrong last time Upper Bounds ƒ Edmundson-Madansky Using bounds in the LShaped Algorithm March 5, 2003 Stochastic Programming – Lecture 15 Slide 2
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Review Why do we care about bounds in stochastic programming? What's “wrong” with numerical integration for evaluating Q ( x ) ? What is Jensen's inequality? How is Jensen's Inequality used in stochastic programming? March 5, 2003 Stochastic Programming – Lecture 15 Slide 3
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Jensen's Inequality in Stochastic Programming E ω [ Q x,ω )] Q x, E ω [ ω ]) We get a tight lower bound on Q x ) by evauating Q x, ¯ ω ) . In our proof, we only used the fact that Q x,ω ) was convex on Ω . So, in general, if φ is a convex function ω of a random variable over its support Ω , then E ω φ ( ω ) φ ( E ω ( ω )) March 5, 2003 Stochastic Programming – Lecture 15 Slide 4
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A Recourse Formulation minimize Q ( x 1 ,x 2 ) = x 1 + x 2 + 5 Z 4 ω 1 =1 Z 2 / 3 ω 2 =1 / 3 y 1 ( ω 1 2 ) + y 2 ( ω 1 2 ) 1 2 subject to ω 1 x 1 + x 2 + y 1 ( ω 1
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  • Spring '08
  • Linderoth
  • Convex function, Jensen's inequality, Upper and lower bounds, stochastic programming, Programming – Lecture

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