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

Branching

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