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6.2 Optimization Constraints by Modulating the Cost Function 77

reoccurs except for the middle region and the right side, close to the last
reference point.
First, we want to note that the problem can be detected here by mon-
itoring the residual distance dist (respectively the cost function E s )
which is here the horizontal distance of the found completion (close to w )
and the input x.
Second, this problem can be solved by re-starting the search: suitable
restart points are the distance-ranked list of node locations a found in the
first place (e.g. the 10th points probes the start locations at node 3,2,1,4).
The procedure stops, if a satisfying solution (low residual cost function)
or a maximum number of trials is reached. Fig. 6.1b demonstrates this
multi-start procedure and depicts the correctly found solutions.
In case the task is known to involve a consecutive sequence of query
points, it is perfectly reasonable to place the previous best-match location

s at the head position of the list of start locations.
Furthermore, the multi-start technique is also applicable to find mul-
tiple best-match solutions. However, extra effort is required to find the
complete list of compatible solutions. E.g. in the middle region of the de-
picted example Fig. 6.1, at least two of the three solutions will be found.

6.2 Optimization Constraints by Modulating the

Cost Function

The best-match minimization procedure Eq. 4.4 can be employed to con-
trol the optimization mechanism in various directions. The multi-start
technique introduced before approaches suitable solutions from different

starting locations s , but using a constant cost function Eq. 4.10

d
X
E s dist x w s p k x k w k s

k
An interesting possibility is to modulate this cost function during the
best-match iteration process. This means that the weight factors p k , which
define the distance metric, are varied in order to influence the iteration
path in a desired way. This approach can be considered, e.g. to avoid local
minima with already known locations.