I’m working on a project related to radiotherapy treatment planning.
The mathematical expression is as following
d is an m\times1 dose distribution vector of each voxel,
x is the n\times1\ fluence weighting vector on each radiation beamlet,
A is the precalculated m\times n dose-fluence deposition matrix
The objective can be any function of x. In our case, the objective function is to minimize the norm to the target dose
Where d_p is the prescribe dose
Subject to the following constraints:
LBdose \le Ax \le UBdose
LBdose,UBdose,d,\ x \geq0
Where LBdose and UBdose are the lower bound and upper bound of the dose d
I tried to script it in CVX, but the model will say that it is not feasible.
n=number of beammlets;
minimize( norm(Ax-Dp) );
LBdose <= Ax <= UBdose;
the message showed on MATLAB:
Barrier solved model in 0 iterations and 161.13 seconds
Model is infeasible
Optimal value (cvx_optval): +Inf
For illustration, let’s say m = 2, n = 1, and A is the 2 by 1 matrix [1;2]. There is no x >= 0 for which 59 <= x <= 66 and 59 <= 2*x <= 66; Hence this simple model, for which all elements of A are nonnegative, is infeasible. So certainly a more complicated model having all elements of A nonnegative, could be infeasible, even though the “reason(s)” for the infeasibility might be more subtle than in my simple example.