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

** Case study background and problem formulations**

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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

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.