This preview shows page 1. Sign up to view the full content.
Unformatted text preview: (¯) = 0.
iix
condition (3a) j =1 (3b) 2014W Econ 172B Operations Research (B) Alexis Akira Toda Proof. By Theorem 2, −∇f (¯) ∈ NC (¯) = (TC (¯))∗ . By Proposition 3 and
x
x
x
the assumption LC (¯) ⊂ co TC (¯), we get LC (¯) = co TC (¯). Hence by the
x
x
x
x
property of dual cones, we get (TC (¯))∗ = (co TC (¯))∗ = (LC (¯))∗ . Now let
x
x
x
J
K be the polyhedral cone generated by {∇gi (¯)}i∈I (¯) and {±∇hj (¯)}j =1 . By
x
x
x
∗
Farkas’s lemma, K is equal to
y ∈ RN (∀i ∈ I (¯)) ⟨∇gi (¯), y ⟩ ≤ 0, (∀j ) ⟨±∇hj (¯), y ⟩ ≤ 0 ,
x
x
x
which is precisely the linearizing cone LC (¯). Again by Proposition Farkas’s
x
lemma, we have (LC (¯))∗ = K . Therefore −∇f (¯) ∈ K , so there exists numbers
x
x
λi ≥ 0 (i ∈ I (¯)) and αj , βj ≥ 0 such that
x
J −∇f (¯) =
x (αj − βj )∇hj (¯).
x λi ∇gi (¯) +
x
j =1 i∈I (¯)
x Letting λi = 0 for i ∈ I (¯) and µj = αj − βj , we get (3a). Finally, (3b) holds
/x
for i ∈ I (¯) since gi (¯) = 0. It also holds for i ∈ I (¯) since we deﬁned λi = 0
x
x
x
for such i.
Deﬁne the Lagrangian of the minimization problem 2 by
J I µj hj (x). λi gi (x) + L(x, λ, µ) = f (x) + j =1 i=1 Then (3a) implies that the derivative of L(·, λ, µ) at x is zero. (3a) is called the
¯
ﬁrstorder condition. (3b) is called the complementary slackness condition. (3a)
and (3b) are jointly called KarushKuhnTucker (KKT) conditions. 4 Constraint qualiﬁcations Conditions of the form LC (¯) ⊂ co TC (¯) in Theorem 4 are called constraint
x
x
qualiﬁcations (CQ). These are necessary conditions in order for the KKT conditions to hold. There are many constraint qualiﬁcations in the literature:
Guignard (GCQ) LC (¯) ⊂ co TC (¯).
x
x
Abadie (ACQ) LC (¯) ⊂ TC (¯).
x
x
J MangasarianFromovitz (MFCQ) {∇hj (¯)}j =1 are linearly independent,
x
and there exists y ∈ RN such that ⟨∇gi (¯), y ⟩ < 0 for all i ∈ I (¯) and
x
x
⟨∇hj (¯), y ⟩ = 0 for all j .
x
Slater (SCQ) gi ’s are convex, hj ’s are aﬃne, and there exists x0 ∈ RN such
that gi (x0 ) < 0 for all i and hj (x0 ) = 0 for all j .
J Linear independence (LICQ) {∇gi (¯)}i∈I (¯) and {∇hj (¯)}j =1 are linearly
x
x
x
independent.
Theorem 5. The following is true for constraint qualiﬁcations.
LICQ or SCQ =⇒ MFCQ =⇒ ACQ =⇒ GCQ. 2014W Econ 172B Operations Research (B) Alexis Akira Toda Proof. ACQ =⇒ GCQ is trivial. The rest is not so simple, so I omit.
In applicat...
View
Full
Document
This document was uploaded on 02/18/2014 for the course ECON 172b at UCSD.
 Winter '08
 Foster,C

Click to edit the document details