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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.

**PROBLEM 1: problem_utility**

Minimize Avg_max_risk (minimizing average of maximum of random linear functions)

subject to

linear ≤ 1 (budget constraint on sum of variables)

Box constraints (variables are not negative)

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Avg_max_risk = Average Max Risk for Loss

Box constraints = constraints on individual decision variables

——————————————————————– Download Problem Data

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Problem statement | Data | Solution | 500 | 10,000 | -7.92264 | 5.59 |

Dataset2 | Problem statement | Data | Solution | 1000 | 10,000 | -5.94599 | 6.6 |

Dataset3 | Problem statement | Data | Solution | 2000 | 10,000 | -7.27767 | 14.99 |

Dataset4 | Problem statement | Data | Solution | 5000 | 10,000 | -5.61194 | 37.61 |

**NOTE:** Problem statements can be simplified using a set of matrices.

**PROBLEM2 : problem_utility-exact**

Minimize Linear + Pm_pen_ni_1 + … + Pm_pen_ni_M (minimizing linear plus sum of partial moments)

subject to

linear ≤ 1 (budget constraint on sum of variables)

Box constraints (variables are not negative)

——————————————————————–

Pm_pen_ni = Partial Moment Penalty for Loss Normal Independent

Box constraints = constraints on individual decision variables

——————————————————————–

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset | 500 | N/A | -7.92114 | 0.04 | |||
---|---|---|---|---|---|---|---|

Environments |
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Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab Subroutines | Matlab Code | Data | |||||

R | R Code | Data |

Instructions for importing problems from Run-File to PSG MATLAB.

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset2 | Problem statement | Data | Solution | 1000 | N/A | -5.94396 | 0.16 |

Dataset3 | Problem statement | Data | Solution | 2000 | N/A | -7.27511 | 1.01 |

Dataset4 | Problem statement | Data | Solution | 5000 | N/A | -5.60658 | 5.98 |

**NOTE:** Problem statements can be simplified using InnerProduct.

**CASE STUDY SUMMARY**

This case study solves Stochastic Utility (or Expected Utility) Problem which is approximated by sampling stochastic parameters of this problem (Sampling Average Approximation approach). The problem formulation and data are based on the dataset considered in Nemirovski et al. (2009). The optimization problem is solved in the approximation format with scenarios (PROBLEM 1: problem_utility) and in the format using normally distributed random variables (PROBLEM 2: problem_utility_exact). The Case Study presents four solved problem instances with 500, 1000, 2000, 5000 variables. ).

**References**

• Nemirovski A., Juditsky A., Lan G. and A. Shapiro (2009): Robust stochastic approximation approach to stochastic programming, SIAM J. Optim., Vol. 19, No. 4, 1574-1609.