Stochastic Optimization ESI 6912

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Stochastic OptimizationESI 6912
NOTES 1 INTRODUCTION

• Instructor Prof. S. Uryasev

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Outline of the course

• Introduction and stochastic programming problem
  formulations
• Two stage problems with recourse continuous
  distributions
• VaR and CVaR problem
• Probabilistic or chance constraint optimization
• Two stage problem discrete distributions

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Applications of Stochastic Programming

• Transportation (uncertain demand)
• Energy Industry (uncertain prices of energy and
  demand)
• Finance (uncertain demand and prices of financial
  instruments)
• Nuclear Engineering (probability of accident)
• Environment (probability of satisfying
  regulations)
• Inventory problem - various areas (uncertain
  demand)
• Classification
• Others (give examples)

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Basic concepts
Stochastic functions - - decision
vector - uncertain variable Loss,
reward, demand, ... Example - number
of units bought - uncertain price
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Basic concepts (contd)
Performance functions
- expected value
-
variance
- mean absolute deviation
- lower
partial moment, where
,
and is a constant value .
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Basic concepts (contd)
Performance functions
- probability
- percentile
(quantile,
Value-at-Risk)
- - Conditional Value-at-Risk
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Expected value, Probability
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? -VaR and ?-CVaR
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Stochastic Programming Problems
deterministic constraints
Example1
Example2
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Newsboy Problem
A news vendor goes to the publisher every
morning and buys x newspapers at a price of c
per paper. The vendor then walks along the
streets to sell as many newspapers as possible at
the selling price r. Any unsold newspaper can be
returned to the publisher at a return price s
(salvage price), with sltc. The news vendor needs
to decide how many newspapers to buy every
morning. Demand for newspapers varies over days
and is described by a random variable
. Assumption the news vendor cannot return to
the publisher during the day to buy more
newspapers
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Newsboy Problem (contd)
c - purchase price r - selling price s - salvage
price x - quantity purchased (decision variable)
- demand (uncertain variable)
Sample performance function (profit)
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Newsboy Problem (contd)
Distribution for demand (discrete distribution)
Expected profit
Losses
Risk functions
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Newsboy Problem (contd)
Maximizing expected profit while controlling the
risk
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1.
Minimizing the risk while controlling expected
profit
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4.