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)nsubject tonlinear ≤ 1 (budget constraint on sum of variables) Box constraints (variables are not negative) ——————————————————————– Avg_max_risk = Average Max Risk for Loss
Box constraints = constraints on individual decision variables ——————————————————————– nDownload Problem Datan

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)nsubject tonlinear ≤ 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 ——————————————————————– 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 SUMMARYn 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.