Biomedical Engineering Reference
In-Depth Information
Some issues in mathematical optimization
I want now to draw your attention to a number of issues that arise in
the context of optimization. Optimization appears on the face of it to
be an automated process. However, in fact, it often needs considerrable
hand-tuning, specific to the application, to make it work well.
Starting Values
In principle, an important insight into the search procedure can be
gained by observing the behavior of the algorithm when one
changes
the starting point
and restarts the search.
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An ideal search process
would arrive at the same result, or at least a clinically comparable
result, regardless of where one started it. If this happens, one can be
relatively content. If any reiteration of the search yields a substantial
improvement over the best previous result, then it has “hopped over”
an intermediate hill and dropped into a lower valley. There is some
chance that this new valley contains the global minimum. When
using searching algorithms, several searches, each using a different
starting value, should be attempted and their results compared.
However, this is rarely done in practice, both because of time
constraints and because the solutions found using the initial set of
starting values are often satisfactory, even if not optimal.
Scale
There is a problem of “scale.” Many treatment variables, such as
distance, angle, intensity etc., have different units and quite different
ranges - e.g., 0 to 20cm for a collimator setting, 0-360
for a gantry
angle, 0 to 2 Gy for a pencil beam weighting factor, and so forth. The
size scale for a step has to be established independently for each
direction. For example, one might pick step sizes of 3 mm for
collimator changes, 5
o
for gantry angle setting, etc. The problem of
scale is also evident in establishing the extent of the search space.
Within what spread of values of each variable will one pick a starting
point? If that spread is too small, and step sizes are too small, it
might not be possible to reach an extremum in a reasonable period of
°
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The user often is unaware of what starting values are used in the optimization
program being used. In the early days of iterative optimization, the analytic
inverse optimization result was sometimes used for the starting values.
