This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Professor John Nachbar Econ 4111 February 14, 2007 Finite Dimensional Optimization, Part II Second Order/Sufficient Conditions 1 Introduction As noted in part I of these notes, the KuhnTucker theorem only gives necessary conditions for a local maximum. Even if one can find an x * and associated KT multipliers satisfying KT, x * need not be even a local maximum, let alone a global maximum. The simplest conditions guaranteeing that the KT conditions are sufficient as well as necessary for a maximum are that the constraint set C is convex and the objective function f is concave. A sufficient condition for C to be convex, in turn, is for the g k to be convex. In these notes I develop these conditions as well as variants relying on weaker conditions (notably quasiconcavity). I do this largely for completeness. The simple rule, KT is sufficient as well as necessary if f concave and the g k are convex, is adequate for most maximization problems that we encounter. 2 Interior Local Optima. The following theorem gives sufficient conditions for a point x * to be a local maxi mum. The theorem does not require that x * be interior. The condition Df ( x * ) = 0 typically wont be satisfied, however, unless x * is interior. Theorem 1. Let f : C R be differentiable, where C R N . Given x * C , suppose that C is locally convex (there is an > such that N ( x * ) C is convex). Suppose that the following conditions hold. 1. Df ( x * ) = 0 . 2. f is locally concave at x * . Then x * is a local maximum. Moreover, if f is locally strictly concave then x * is a local strict maximum. Proof. The proof is by contraposition. Suppose that f is locally concave at x * but that x * is not a local maximum. I show that Df ( x * ) 6 = 0. Choose any > such that f is concave on N ( x * ). Since x * is not a local maximum, there is an 1 x N ( x * ) such that f ( x ) > f ( x * ). Take any (0 , 1) and let x = x +(1 ) x * . By concavity, f ( x ) f ( x ) + (1 ) f ( x * ) = f ( x * ) + ( f ( x ) f ( x * )) . Rewriting, noting that x = x * + ( x * x ), and dividing by (which is positive) yields, f ( x * + ( x * x )) f ( x * ) = f ( x ) f ( x * ) ( f ( x ) f ( x * )) = f ( x ) f ( x * ) > . Taking the limit as 0, this implies that Df ( x * )[ x * x ] > 0, so that Df ( x * ) 6 = 0. (See the notes on differentiation.) On the other hand, suppose that x * is not a local strict maximum. Again, choose any > 0 such that f is strictly concave on N ( x * ). Since x * is not a local strict maximum, there is an x N ( x * ) such that f ( x ) f ( x * ). Take any (0 , 1) and let x = x +(1 ) x * . By strict concavity, f ( x ) > f ( x )+(1 ) f ( x * ). Rewriting yields f ( x * + ( x * x )) f ( x * ) = f ( x ) f ( x * ) > ( f ( x ) f ( x * )) = f ( x ) f ( x * ) ....
View
Full
Document
This note was uploaded on 05/20/2010 for the course ECON 511 taught by Professor Johnnachbar during the Fall '08 term at Washington University in St. Louis.
 Fall '08
 JohnNachbar

Click to edit the document details