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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 Kuhn-Tucker 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 quasi-concavity). 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 * ) ....
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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