Optimum Design
167
VIII. GEOMETRIC PROGRAMMING
First let’s look at some inequalities [3.3,3.8,3.25]
(U 1 -- e2) 2 >_ 0
Vl ~ - 2vl ~2 + ~2
add 4U1 U2 to both sides
U~ 2 + 2U1 U2 + U2 ~ > 4U~U2
Now take the square root
UI + U2 ~ 2U~/.
2
Divide by 2 yields
1
1 _ ~/2~/2
~ gl
+~ U2 > ~1
~2
consider four non-negative numbers
~~1111
(U~+U~)’/~(U~+U4)
~/~2
2
~gl +~g~ +~g~ +~g4 ~ .
-
-
On the right hand side use Eq. (3.58)
gl + U2 ~ 2U~/2 U~/2
U3 + U4 e 2u~/gu~/2
So Eq. (3.59) becomes
---U~-I
1
1
1 -- (l[l121[l/2~l12glll12[[l121112e
4 Ul + 4 U2 + 4-o + 4 U4 > ,~l ~2 ,~ ~4
Zener made the observation if one lets
and
ui = 6iUi then Eq. (3.60)
(3.57)
(3.58)
(3.59)
(3.60)
(3.61)
(3.62)