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The sound of sediments : acoustic sensing in uncertain environments
van Leijen, A.V.Publication date 2010
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Citation for published version (APA):
van Leijen, A. V. (2010). The sound of sediments : acoustic sensing in uncertain environments.
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Appendix C
Test functions for tuning
In chapter 7, metaheuristic optimization schemes have been tuned using more or less representative test functions. These test functions have been derived from an original function Φ6 that was constructed by Fallat and Dosso [32] as:
Φ6(x1, ..., x6) = 4.8 + x21 + 5x22 + 0.1x23 + 0.05x24 + x25 + x26 − 0.3 cos(4π(x1 − x2)) − 1.4 cos(4π(x1 + x2)) − 0.5 cos(10π(0.05x4 − 0.1x3)) − 1.0 cos(10π(0.05x4 + 0.1x3)) − 0.25 cos(5π(x5 − x6)) − 1.35 cos(5π(x5 + x6)) . (C.1)
By construction, Φ6(0, ..., 0) = 0 with (0, ..., 0) as the global minimum and only
zero-solution. Conversely, the considered inverse problems with real data have ob-jective functions that are strictly positive. The metaheuristics are therefore tuned using similar but strictly positive functions Φ4 and Φ8. These fast and
represen-tative test functions have four or eight parameters and are both constructed with 111
112 Appendix C. Test functions for tuning
a minimum of 1.5, and are defined as
Φ4(x1, .., x4) = 4.8 + x21 + 5x22 + x23 + x24 − 0.3 cos(4π(x1 − x2)) − 1.4 cos(4π(x1 + x2)) − 0.25 cos(5π(x3 − x4)) − 1.35 cos(5π(x3 + x4)) (C.2) and Φ8(x1, .., x8) = 7.85 + x21 + 5x22 + 0.1x23 + 0.05x24 + x25 + x26 + 0.5x27 + 2x28 − 0.3 cos(4π(x1 − x2)) − 1.4 cos(4π(x1 + x2)) − 0.5 cos(10π(0.05x4 − 0.1x3)) − 1.0 cos(10π(0.05x4 + 0.1x3)) − 0.25 cos(5π(x5 − x6)) − 1.35 cos(5π(x5 + x6)) − 0.35 cos(7π(x7 − x8)) − 1.2 cos(7π(x7 + x8)) . (C.3)