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140 Gradient Optimization
condition for a minimum. By similar triangles,
G5-GI 0,
G5 = a . (5.64)
The increase beyond the 0 step just taken is a- 0" so that
- GI 5
a-o,=o,Z, where Z- G5-GI . (5.6 )
Fletcher limits the extrapolation to be no more than four times the prior step;
Le., Z in (5.65) is limited to 4. The variable names employed correspond to the
program code to follow.
Figure 5.20 shows that a minimum has been bounded in 0 when either
F(o) > F i or when the slope is positive. Suppose that this occurs at 0 = >-.. There
are now four pieces of information: the two function values, F(O) = F and
F(>-')=F9; and the two slopes F(0)=G5 and F(>-')=GI. These four items
enable the fit of a cubic function, which can interpolate the minimum between
the bounds. The cubic function approximates a flat spring fitted to the known
function values and slopes, provided that the slopes are small. Davidon (1959)
suggested the following formulation, and it has been widely applied since then.
Suppose that the fitting function has the form
Then, at a=;\,
h(a)=ao+a,0+a20 2 +a30 3 . (5.66)
F9= F +G5 ·>-.+a2>-.2+ a3>-.3,
GI =G5+ 2a 2 >-.+ 3a 3 >-.2.
The last two equations can be solved for coefficients a 2 and a 3 :
3(F9-F) ->-.(205 +GI)
a2=
>-.2
2(F-F9)H(G5+Gl)
a3=
>-.3
It is convenient to define the constant z as
3(F- F9)
z >-. +GI+G5.
(5.67)
(5.68)
(5.69)
(5.70)
(5.71 )
The cubic interpolation step in 0 is then obtained by differentiating (5.66) and
equating that to zero. The root of the resulting equation that is between 0 = 0
and 0 =>-. is thus obtained after considerable algebra:
A I-(GI+W-z)
0=>-' 2W+GI-G5 '
where an additional defined constant is
(5.72)
W=(z2-G5XGlj'/2. (5.73)
The forms of these equations are designed to minimize cancellation by