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- To minimise f(x),
subject to the constraint g(x) = c,
consider
- Λ(x, λ) = f(x) + λ { g(x) - c }.
- λ is known as the "Lagrange multiplier."
-
- Solve
- ∇ Λ(x, λ)
= (∂Λ/∂x1, ..., ∂Λ/∂xn, ∂Λ/∂λ)
= 0
-
- For example,
given positive integers
{n1, ..., nk},
minimise
- n1 log p1 + ... + nk log pk
- subject to
- p1 + ... + pk = 1,
- let
- Λ(p, λ)
= n1 log p1 + ...
+ nk log pk
+ λ{p1 + ... + pk - 1},
- ∇ Λ =
(n1/p1 + λ,
...,
nk/pk + λ,
p1 + ... + pk - 1)
= 0,
- so
- pi = - ni / λ ∝ ni
(λ can be negative)
- and
- ∑ pi = 1,
- giving
- pi = ni / ∑j nj
(and λ = - ∑j nj).
- (Also see the
[multinomial]
probability distribution.)
-
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