# Case Study: Distribution Approximation by Maximizing Entropy with Second-Order Stochastic Dominance and Moment Constraints

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Case study background and problem formulations

Instructions for optimization with PSG Run-File, PSG MATLAB Toolbox, PSG MATLAB Subroutines and PSG R.

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Problem:
Maximize -Entropyr (Relative Entropy)
subject to
pCvar ≥ Vector (second-order stochastic dominance constraints)
linear = Const1 (average constraint)
linear ≤ Const2 (moment constraint)
linear = 1 (probability constraint)
Box constraints (nonnegative probabilities)
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Entropyr = Relative Entropy
pCvar = CVaR for Discrete Distribution as a Function of Atom Probabilities
Linear = Linear function
Box constraints = constraints on individual decision variables

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PSG Run-File

Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 3.50GHz (sec)
Dataset1 Problem Statement Data Solution 100 9 4.519554668226 0.58
Dataset2 Problem Statement Data Solution 1000 99 6.907578290799 3.39
Dataset3 Problem Statement Data Solution 4000 399 8.294036180882 114.19

PSG MATLAB Subroutines

Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 3.50GHz (sec)
Dataset1 Matlab code Solution 100 9 4.519561179885 0.22

CASE STUDY SUMMARY
This case study is based on the methodology developed by Mafusalov (2014). We approximated a given distribution with a new distribution in a conservative way. Left- and right- tail conditional expectations are considered, to make sure that the tail behavior of the given distribution is conservatively represented in the new distribution. The estimation problem is formalized as an entropy maximization with the Second-Order Stochastic Dominance (SOSD) constraints and the second moment constraint. The SOSD constraints are reduced to linear constrains. The case study showed that PSG can solve quite efficiently the considered optimization problems with various sizes of datasets.

References
• Mafusalov A. (2014). Entropy Maximization with Stochastic Dominance and Moment Constraints for Distribution Approximation. Presentation at INFORMS Annual Meeting, San Francisco, November 2014, and at “Risk Management Approaches in Engineering Applications” workshop, Gainesville, November 2014.