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

PROBLEM: problem_men
Minimizing Pm2_pen_g (minimizing partial moment two penalty function)
subject to
Box constraints (bounds on variables)
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Pm2_pen_g = Partial Moment Two Penalty for Gain
Box constraints = constraints on individual decision variables
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Data and solution in Run-File Environment

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 2.66GHz (sec)
Dataset1	Problem Statement	Data	Solution	10	10	0.00035009773	<0.01
Dataset2	Problem Statement	Data	Solution	1,113	20,859	15.490633332895	312.76
Dataset3	Problem Statement	Data	Solution	2,736	44,362	5.379941523030	879.34
Dataset4	Problem Statement	Data	Solution	2,055	37,794	3.875103565617	103.19

Data and solution in MATLAB Environment

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.50GHz (sec)
Dataset1	Matlab code	Data	Solution	10	10	0.000350098	<0.01
Dataset2	Matlab code	Data	Solution	1,113	20,859	15.4906	306.52
Dataset3	Matlab code	Data	Solution	2,736	44,362	5.37994	864.84
Dataset4	Matlab code	Data	Solution	2,055	37,794	3.87768	91.27

Data and solution in R Environment

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.50GHz (sec)
Dataset1	R code	Data		10	10	0.000350098	<0.01

CASE STUDY SUMMARY
This case study solves an intensity-modulated radiation therapy (IMRT) treatment planning problem. The problem statement is presented in the paper “An exact approach to direct aperture optimization in IMRT treatment planning” by Men et al., (2007). In the original case study, the sum of squares of penalties is minimized to reduce the radiation therapy damage. PSG uses the “partial moment two penalty” (pm2_pen) function, which is the average of sum of squares of penalties. Therefore, the optimal point obtained with PSG is the same as in the original case study; however, the PSG objective value differs from the original objective value by a fixed multiplier.
The scenarios matrices in radiation therapy case studies are sparse (few non-zero elements). Therefore, packed matrix format (pmatrix) is quite beneficial for these problems. We solve four problems which differ only by the dataset (all problems have the same mathematical formulation). Dataset1 has no connections with real life problems: it is demonstrative, the rest three dataset are of the big size and represent real life problems.

Therefore problem is solved using 4 datasets:
• Dataset1 for “short case study” including 10 variables and matrix of scenarios with 10 scenarios;
• Dataset2 including 1,113 variables and matrix of scenarios with 20,859 scenarios;
• Dataset3 including 2,736 variables and matrix of scenarios with 44,362 scenarios;
• Dataset4 including 2,055 variables and matrix of scenarios with 37,794 scenarios.

References
• Men, C., Romeijn, E., Taskin, C. and J. Dempsey, (2007): An Exact Approach to Direct Aperture Optimization in IMRT Treatment Planning, in Physics in Medicine and Biology, Vol. 52 , pp. 7333–7352.