Lecture 4

# Lecture 4

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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 ¯ ﬁrst-order condition. (3b) is called the complementary slackness condition. (3a) and (3b) are jointly called Karush-Kuhn-Tucker (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 Mangasarian-Fromovitz (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...
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## This document was uploaded on 02/18/2014 for the course ECON 172b at UCSD.

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