diapositiva 1 - ntnu
TRANSCRIPT
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Math Background
Antonio J. Conejo
Univ. Castilla – La Mancha
20111
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Contents
— Complementarity
— Stochastic programming
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Complementarity
— Optimization problem
— OPcOP
—MPEC
—Equilibrium
—EPEC
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References• R. W. Cottle, J.-S. Pang, and R. E. Stone. The Linear
Complementarity Problem. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2009.
• Z.-Q. Luo, J.-S. Pang, and D. Ralph. Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge, UK, 2008.
• S. A. Gabriel, A. J. Conejo, J. D. Fuller, B. F. Hobbs, C. Ruiz Complementarity Modeling in Energy Markets Springer, New York, Fall 2011.
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Complementarity(Optimization Problem)
5
Objective function (minimize or maximize)
Optimization Problem (OP)
subject to:
Equality constraints
Inequality constraints
Bound on variables
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Complementarity(MPEC)
6
subject to:
Mathematical Program with Equilibrium Constraints
(MPEC)
Constraining optimization problem 1
Constraining optimization problem n
Objective function (minimize or maximize)
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Complementarity(MPEC-KKT)
7
subject to:
KKT conditions of constraining problem 1
KKT conditions of constraining problem n
Objective function (minimize or maximize)
Mathematical Program with Equilibrium Constraints
(MPEC)
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Complementarity(Equilibrium)
8
Optimization
problem nOptimization
problem 1
Equilibrium Problem (EP)
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Complementarity(Equilibrium-KKT)
9
KKT
conditions of
problem 1
KKT
conditions of
problem n
Equilibrium Problem (EP)
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Complementarity(EPEC)
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MPEC 1
MPEC n
Equilibrium Problem with Equilibrium Constraints
(EPEC)
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Stochastic Programming
— References
— Two-stage stochastic programming
— Decision framework
— Decision tree
— Scenario formulation
— Node formulation
— EVPI
— VSS
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References
— J. R. Birge and F. Louveaux. “Introduction to StochasticProgramming”. Springer, New York, New York, 1997.
— J. L. Higle. Tutorials in Operations Research, INFORMS 2005.Chapter 2: “Stochastic Programming: Optimization WhenUncertainty Matters”. INFORMS, Hanover, Maryland, 2005.
— A. J. Conejo, M. Carrión, J. M. Morales, “Decision MakingUnder Uncertainty in Electricity Markets” International Seriesin Operations Research & Management Science, Springer,New York. 2010.
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Two-stage stochastic programming
0,y,xg
0,y,xh to subject
,y,xfO minimizex,y
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Expectation CVaR
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Decision framework
— Decisions x are made
— Stochastic vector λ realizes in a scenario
— Given x, decisions y(x,λ) are made for each realization of λ
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Decision tree
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Decision tree
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Scenario formulation
low
average
high
3 3
22
11
17
Realization
Probability
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Scenario formulation
xxx
3,2,1i 0,y,xg
3,2,1i 0,y,xh to subject
,y,xf Z minimize
321
iii
iii
iii
3
1i
iy,y,y,x,x,x
S
321321
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Non-anticipativity constraint
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321 xxx
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Node formulation
20
3,2,1i 0,y,xg
3,2,1i 0,y,xh to subject
,y,xfZ minimize
ii
ii
ii
3
1i
i
S
y,y,y,x 321
Non-anticipativity constraints are implicit!
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EVPI
EXPECTED
VALUE of the
PERFECT
INFORMATION
Measure of the value of “perfect” information
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EVPI
22
3,2,1i 0,y,xg
3,2,1i 0,y,xf to subject
,y,xf Z minimize
iii
iii
iii
3
1i
i
P
y,y,y,x,x,x 321321
No anticipativity constraints!
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EVPI
23
PS ZZEVPI
EVPI is non-negative
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VSS(only expectation)
VALUE of the
STOCHASTIC
SOLUTION
Measure of the relevance (gain) of using astochastic approach
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VSS
25
D
avg
avg
avg
yx,
x Solution
0,y,xg
0,y,xh to subject
,y,xf maximize
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VSS
26
3
1i
ii
D
i
D ,y,xfZ
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VSS
27
SD ZZ VSS
VSS is non-negative