My function is nonlinear with respect to a scalar \alpha .
However, the calculation of objective function is very time consuming, making optimization also very time consuming. Also, I have to do it for 1/2 millon voxels (3d equivalent of pixels). I plan to do it using “lsqnonlin” of matlab.
Rather than optimizing over all possible real values, I plan to search over preselected 60 values. My variable \alpha (or flip angle error) could be anything between 0-35%; but, I want to pass only linearly spaced points as candidates (i.e. 0:005:0.35). In other words, I want lsqnonlin to choose possible solution only from (0:005:0.35). Since I can pre-calculate objective values for these, it would be very fast. In other words, I need to restrict search space.
Here, I am talking about single voxel; though I performs lsqnonlin over multivoxel and corresponding \alpha is mapped accordingly to a column vector.
I can not do grid search over preselected value as I plan to perform spatial smoothing in 3D. Some guidance would be highly appreciated.
Regards, Dushyant
Hi Dushyant:
I'd think about reformulating your problem in terms of mixed integer programming, either linear or non-linear and then using some exact (e.g. branch and cut) or heuristic (e.g., SA) method. However, this may require some more effort to formulate and code the problem.
My two cents to the question:
1) What you want to do is the discrete optimization (see the link). The discrete coded (binary) genetic algorithm, e.g., could do the trick.
2) You could perform the factorial design of experiments (factorial d.o.e.) with a number of levels for each parameter you want, but the fraction should be very small to calculate it (see below, why).
1/2 millon voxels as optimization variables is somewhat difficult to optimize, even assuming varying them on 60 levels. It is probably possible to run on supercomputer, but first consider if the problem is formulated correctly and the results would be significant, if it is worth trying.
https://en.wikipedia.org/wiki/Discrete_optimization
The easiest way to confine the solution space considered is to transform the variables so that all values map into the desired space. I would then sample the space randomly and hope that a pattern emerges.
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