MILP Formulationsfor Stochastic Dominance
JamesLuedtkeNew Formulations for Optimization Under Stochastic Dominance ConstraintsSIAM Journal on Optimization, 2008
Stochastic dominance
Recall:
So, for the case
The problems
No assumptions neitheron X nor g
A simplification
We can focus on
with
and assume:
Second-order stochastic dominance
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A characterization
SDLP
constraintsvariables
Another characterization
and Y with finite means.
(Strassen)
cSSD1
variables
constraints
(Luedtke)
cSSD2
We can actually replace with
Better performance in practice(using CPLEX dual simplex)
First-order stochastic dominance
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A characterization
FDMIP
variables
constraintsPoor LP
relaxation bounds
cFSD
variables
constraints
(Luedtke)
FSD
The LP relaxationyields a formulationfor SSD
Solving the MILP
If: ● X is a polyhedron and ● g(x, \xi^iare affine in x
is a MILP
Branching
Select levelyi should begreater than
At most one of thesecan be positive
SOS1
Order preserving heuristic
Sort
Fix so it satisfies:
Solve
Computational results
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Portfolio optimization
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 of100000 secs
First-order dominance (root LP)
FDMIP: Before addingCPLEX cuts FDMIP.C: After addingCPLEX cuts
FDMIP
FDMIP.C
cFSD
First-order stochastic dominance
H: Heuristics B: Specialized branching