Evolution and Optimum Seeking

36 Hill climbing Strategies
parabola as the model function (quadratic interpolation). Assuming that the three points
are a (k) < b (k) < c (k) , with the objective function values F (a (k) )F(b (k) ) and F (c (k) ),
the trial parabola P (x) hasavanishing rst derivative atthepoint
d (k) = 1
2
[(b (k) ) 2 ; (c (k) ) 2 ] F (a (k) )+[(c (k) ) 2 ; (a (k) ) 2 ] F (b (k) )+[(a (k) ) 2 ; (b (k) ) 2 ] F (c (k) )
[b (k) ; c (k) ] F (a (k) )+[c (k) ; a (k) ] F (b (k) )+[a (k) ; b (k) ] F (c (k) )
(3.16)
This point is a minimum only if the denominator is positive. Otherwise d (k) represents
a maximum or a saddle point. In the case of a minimum, d (k) is introduced as a new
argument value and one of the old ones is deleted:
a (k+1) = a (k)
b (k+1) = d (k)
c (k+1) = b (k)
a (k+1) = d (k)
b (k+1) = b (k)
c (k+1) = c (k)
a (k+1) = b (k)
b (k+1) = d (k)
c (k+1) = c (k)
a (k+1) = a (k)
b (k+1) = b (k)
c (k+1) = d (k)
9
>=
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9
>=
>
9
>=
>
9
>=
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( if a (k)