Stochastic dominance

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<ul><li><p>MILP Formulationsfor Stochastic Dominance</p></li><li><p>JamesLuedtkeNew Formulations for Optimization Under Stochastic Dominance ConstraintsSIAM Journal on Optimization, 2008</p></li><li><p>Stochastic dominance</p><p>Recall:</p><p>So, for the case </p></li><li><p>The problems</p><p>No assumptions neitheron X nor g</p></li><li><p>A simplification</p><p>We can focus on</p><p>with</p><p>and assume:</p></li><li><p>Second-order stochastic dominance</p><p>Image: http://www.flickr.com/photos/admiriam/</p></li><li><p>A characterization</p></li><li><p>SDLP</p><p>constraintsvariables</p></li><li><p>Another characterization</p><p> and Y with finite means.(Strassen)</p></li><li><p>cSSD1</p><p>variables</p><p>constraints</p><p>(Luedtke)</p></li><li><p>cSSD2</p><p>We can actually replace with</p><p>Better performance in practice(using CPLEX dual simplex)</p></li><li><p>First-order stochastic dominance</p><p>Image: http://www.flickr.com/photos/kome8/</p></li><li><p>A characterization</p></li><li><p>FDMIP</p><p>variables</p><p>constraintsPoor LP</p><p>relaxation bounds</p></li><li><p>cFSD</p><p>variables</p><p>constraints</p><p>(Luedtke)</p></li><li><p>FSD</p><p>The LP relaxationyields a formulationfor SSD</p></li><li><p>Solving the MILP</p><p>If: X is a polyhedron and g(x, \xi^iare affine in x</p><p>is a MILP</p></li><li><p>Branching</p><p>Select levelyi should begreater than</p><p>At most one of thesecan be positive</p><p>SOS1</p></li><li><p>Branching</p></li><li><p>Order preserving heuristic</p><p>Sort</p><p>Fix so it satisfies:</p><p>Solve</p></li><li><p>Computational results</p><p>Image: http://www.flickr.com/photos/piper/</p></li><li><p>Portfolio optimization</p></li><li><p>Portfolio optimization</p><p> 435 stocks in S&amp;P 500 N daily returns in years 2005, 2006, 2007</p><p> CPLEX 9.0 2.4 GHz, 2GB memory</p></li><li><p>Second-order dominance</p><p>Time limit of100000 secs</p></li><li><p>First-order dominance (root LP)</p><p>FDMIP: Before addingCPLEX cuts FDMIP.C: After addingCPLEX cuts</p><p>FDMIP</p><p>FDMIP.C</p><p>cFSD</p></li><li><p>First-order stochastic dominance</p><p>H: Heuristics B: Specialized branching</p></li></ul>