# Case Study: Stochastic Utility Problem

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
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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
——————————————————————–

Dataset 500 N/A -7.92114 0.04 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data R R Code Data
Download other datasets in Run-File Environment.
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.