Math Background

Antonio J. Conejo

Univ. Castilla – La Mancha

20111

Contents

— Complementarity

— Stochastic programming

2

Complementarity

— Optimization problem

— OPcOP

—MPEC

—Equilibrium

—EPEC

3

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.

4

Complementarity(Optimization Problem)

5

Objective function (minimize or maximize)

Optimization Problem (OP)

subject to:

Equality constraints

Inequality constraints

Bound on variables

Complementarity(MPEC)

6

subject to:

Mathematical Program with Equilibrium Constraints

(MPEC)

Constraining optimization problem 1

Constraining optimization problem n

Objective function (minimize or maximize)

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)

Complementarity(Equilibrium)

8

Optimization

problem nOptimization

problem 1

Equilibrium Problem (EP)

Complementarity(Equilibrium-KKT)

9

KKT

conditions of

problem 1

KKT

conditions of

problem n

Equilibrium Problem (EP)

Complementarity(EPEC)

10

MPEC 1

MPEC n

Equilibrium Problem with Equilibrium Constraints

(EPEC)

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

13

Expectation CVaR

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

15

Decision tree

16

Scenario formulation

low

average

high

3 3

22

11

17

Realization

Probability

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

18

Non-anticipativity constraint

19

321 xxx

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!

EVPI

EXPECTED

VALUE of the

PERFECT

INFORMATION

Measure of the value of “perfect” information

21

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!

EVPI

23

PS ZZEVPI

EVPI is non-negative

VSS(only expectation)

VALUE of the

STOCHASTIC

SOLUTION

Measure of the relevance (gain) of using astochastic approach

24

VSS

25

D

avg

avg

avg

yx,

x Solution

0,y,xg

0,y,xh to subject

,y,xf maximize

VSS

26

3

1i

ii

D

i

D ,y,xfZ

VSS

27

SD ZZ VSS

VSS is non-negative