Stochastic dominance

Nov 15, 20111 like759 views

The document summarizes formulations for optimization problems under stochastic dominance constraints. It presents formulations for both first-order and second-order stochastic dominance. For second-order stochastic dominance, it introduces the SSDLP and cSSD formulations. For first-order stochastic dominance, it introduces the FDMIP, cFSD, and FSD formulations. It also discusses solving the resulting mixed integer programs using cutting planes and specialized branching heuristics.

MILP
Formulations
for Stochastic Dominance

James
Luedtke
New Formulations for
Optimization Under
Stochastic Dominance
Constraints
SIAM Journal on
Optimization, 2008

Stochastic dominance
Recall:

So, for the case

The problems

No assumptions neither
on X nor g

A simplification

We can focus on

with

and
assume:

Image: http://www.flickr.com/photos/admiriam/

Second-order stochastic dominance

Another characterization
(Strassen)
and Y with finite means.

cSSD1

(Luedtke)

variables

constraints

cSSD2

We can actually replace

with

Better performance in practice
(using CPLEX dual simplex)

Image: http://www.flickr.com/photos/kome8/

First-order stochastic dominance

FDMIP

n ds
b ou
LP ion
variables
or xat
Po la
constraints re

FSD

The LP relaxation
yields a formulation
for SSD

Solving the MILP
If:
�� X is a polyhedron and
�� g(x, xi^iare affine in x

is a MILP

Branching

Select level
yi should be
greater than

At most one of these
can be positive

SOS1

Order preserving heuristic

Fix so it satisfies:

Sort

Solve

Image: http://www.flickr.com/photos/piper/

Computational results

Portfolio optimization

�� 435 stocks in S&P 500
�� N daily returns in years 2005, 2006, 2007

�� CPLEX 9.0
�� 2.4 GHz, 2GB memory

Second-order dominance

Time limit of
100000 secs

First-order dominance (root LP)

FDMIP:
Before adding
CPLEX cuts

FDMIP.C:
After adding
CPLEX cuts

FDMIP

FDMIP.C

cFSD

First-order stochastic dominance

H: Heuristics

B: Specialized
branching

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Stochastic dominance

1. MILP Formulations for Stochastic Dominance

2. James Luedtke New Formulations for Optimization Under Stochastic Dominance Constraints SIAM Journal on Optimization, 2008

3. Stochastic dominance Recall: So, for the case

4. The problems No assumptions neither on X nor g

5. A simplification We can focus on with and assume:

6. Image: http://www.flickr.com/photos/admiriam/ Second-order stochastic dominance

7. A characterization

8. SDLP variables constraints

9. Another characterization (Strassen) and Y with finite means.

10. cSSD1 (Luedtke) variables constraints

11. cSSD2 We can actually replace with Better performance in practice (using CPLEX dual simplex)

12. Image: http://www.flickr.com/photos/kome8/ First-order stochastic dominance

13. A characterization

14. FDMIP n ds b ou LP ion variables or xat Po la constraints re

15. cFSD (Luedtke) variables constraints

16. FSD The LP relaxation yields a formulation for SSD

17. Solving the MILP If: �� X is a polyhedron and �� g(x, xi^iare affine in x is a MILP

18. Branching Select level yi should be greater than At most one of these can be positive SOS1

19. Branching

20. Order preserving heuristic Fix so it satisfies: Sort Solve

21. Image: http://www.flickr.com/photos/piper/ Computational results

22. Portfolio optimization

23. Portfolio optimization �� 435 stocks in S&P 500 �� N daily returns in years 2005, 2006, 2007 �� CPLEX 9.0 �� 2.4 GHz, 2GB memory

24. Second-order dominance Time limit of 100000 secs

25. First-order dominance (root LP) FDMIP: Before adding CPLEX cuts FDMIP.C: After adding CPLEX cuts FDMIP FDMIP.C cFSD

26. First-order stochastic dominance H: Heuristics B: Specialized branching

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Stochastic dominance

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