Sensitivity

Strong Duality > Slater Condition | Optimality conditions | Sensitivity | Applications

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 .

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)