- #1
Trollfaz
- 137
- 14
A simple linear problem goes
min c'x such that f_i(x)<= 0 and Ax=b
x
Suppose we make all constraints affine. Then
Dx-e<=0 and Ax-b =0
We get the Langrangian function as
c'x + λ'(Dx-e) +ν'(Ax-b) and since Ax-b is 0,
we reduce L to
c'x + λ'(Dx-e)
The dual function g is
inf L(x,λ)
x
Then I differentiate L against x to get c=-D'λ
With that we get g(λ) as -λ.e
So I conclude that d*=sup g(λ) against λ.
Does differentiation of the Lagrangian function give me the dual optimum and does it work for all convex inequality constraints and convex objective functions?
min c'x such that f_i(x)<= 0 and Ax=b
x
Suppose we make all constraints affine. Then
Dx-e<=0 and Ax-b =0
We get the Langrangian function as
c'x + λ'(Dx-e) +ν'(Ax-b) and since Ax-b is 0,
we reduce L to
c'x + λ'(Dx-e)
The dual function g is
inf L(x,λ)
x
Then I differentiate L against x to get c=-D'λ
With that we get g(λ) as -λ.e
So I conclude that d*=sup g(λ) against λ.
Does differentiation of the Lagrangian function give me the dual optimum and does it work for all convex inequality constraints and convex objective functions?