|
dHc _ daj (t) |
1 |
+ λ(t) = 0, j = 1, . . . , n2 |
(7) | ||
|
dλ л - ρλ = |
dH |
= 2n1c1x(t) + 2n2c2x(t) + λ(t) |
b- |
2x(t) |
(8) |
|
dHc _ |
a(t) - |
x2(t) |
(9) | ||
From (6) and (7),
λ(t)
1
ai(t)
1
aj(t)
i = 1,...,n1, j = 1,...,n2,
which implies
ai(t) = aj (t) ∀i = 1, . . . , n1 , j = 1, . . . , n2
(10)
(11)
Moreover:
λ(t)
1
ai(t)
i = 1, . . . ,n1 implies n1ai(t) =
ni
λ(t)
and therefore
n1 n2
= n1ai(t), and by similar argument = n2aj (t),
i=1 j=1
which means that
n1 n2
a(t) = ai(t) + aj(t) i = 1,...,n1, j = 1,...,n2 (12)
i=1 j=2
and
a(t) = n1ai(t) + n2aj(t) i = 1, . . . , n1, j = 1, . . . , n2 (13)
Note that λ can be expressed as a function of a via the following reasoning:
λ(t)
1
ai(t)
i = 1, . . . ,n1 and λ(t)
1
aj(t)
j = 1, . . . ,n2
⇒ ai(t)λ(t) = -1 i = 1, . . . , n1 and aj (t)λ(t) = -1 j = 1, . . . , n2
n1 n2
⇒ λ(t) ai(t) = -n1 and λ(t) aj (t) = -n2
i=1 j=1
Adding the two together:
λ(t) ∑ ai(t) + λ(t) ∑ aj (t) = -(n + П2)
i=1 j=1