Basic idea via geometry

Details

Nominal problem:
 p^ast(u,v) := min_x : f_0(x) ~:~ Ax = b+v, ;; f_i(x) le u_i, ;; i=1,ldots, m,
Dual:
 d^ast = max_{lambda ge 0, : nu} : g(lambda,nu).

Perturbed problem:
 p^ast(u,v) := min_x : f_0(x) ~:~ Ax = b+v, ;; f_i(x) le u_i, ;; i=1,ldots, m,
Dual of perturbed problem:
 d^ast = max_{lambda ge 0, : nu} : g(lambda,nu)-lambda^Tu - nu^Tv.
With same function g.

Apply weak duality to perturbed problem:
 p^ast(u,v) ge g(lambda^ast,nu^ast) - u^Tlambda^ast - v^Tnu^ast .
If p^ast(cdot,cdot) is differentiable, then (lambda^ast,nu^ast) is the gradient with respect to (u,v)