A Finding dual optimum of a linear problem

Thread starter Trollfaz
Start date Oct 26, 2023
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Linear programming

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The discussion focuses on finding the dual optimum of a linear problem defined by affine constraints. The Lagrangian function is derived as c'x + λ'(Dx-e) when constraints are transformed. By differentiating the Lagrangian with respect to x, it is established that c = -D'λ. The dual function g(λ) is then expressed as -λ.e, leading to the conclusion that d* equals the supremum of g(λ). The query raised is whether this differentiation method yields the dual optimum for all convex inequality constraints and convex objective functions.

Oct 26, 2023

Trollfaz

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?

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