Background

Problem. Minimizing partial moment two penalty function

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

 

 

Background

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). Problem 1 has no connections with real life problems: it is demonstrative, the rest three dataset are of the big size and represent real life problems.

 

Problem

 

Simplified Problem Statement

 

Minimizing Pm2_pen_g (minimizing partial moment two penalty function)

 subject to

Box constraints (bounds on variables)

 

where

 

Pm2_pen_g = Partial Moment Two Penalty for Gain

Box constraints = constraints on individual decision variables

 

Mathematical Problem Statement

 

 

Problem dimension and solving time

 

 

Solution in Run-File Environment

 

 

Input Files to run CS:

 

Output Files:

 

Solution in MATLAB Environment

 

Solved with riskprog PSG subroutine (General (Text) Format of PSG in MATLAB):

 

Input Files to run CS: