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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 won’t 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 ε > such that N ε ( 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 ε > such that f is concave on N ε ( x * ). Since x * is not a local maximum, there is an 1 ˆ x ∈ N ε ( x * ) such that f (ˆ x ) > f ( x * ). Take any θ ∈ (0 , 1) and let x θ = θ ˆ x +(1- θ ) x * . By concavity, f ( x θ ) ≥ θf (ˆ x ) + (1- θ ) f ( x * ) = f ( x * ) + θ ( f (ˆ x )- f ( x * )) . Rewriting, noting that x θ = x * + θ ( x *- ˆ x ), and dividing by θ (which is positive) yields, f ( x * + θ ( x *- ˆ x ))- f ( x * ) θ = f ( x θ )- f ( x * ) θ ≥ θ ( f (ˆ x )- f ( x * )) θ = f (ˆ x )- f ( x * ) > . Taking the limit as θ → 0, this implies that Df ( x * )[ x *- ˆ 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 ε > 0 such that f is strictly concave on N ε ( x * ). Since x * is not a local strict maximum, there is an ˆ x ∈ N ε ( x * ) such that f (ˆ x ) ≥ f ( x * ). Take any θ ∈ (0 , 1) and let x θ = θ ˆ x +(1- θ ) x * . By strict concavity, f ( x θ ) > θf (ˆ x )+(1- θ ) f ( x * ). Rewriting yields f ( x * + θ ( x *- ˆ x ))- f ( x * ) θ = f ( x θ )- f ( x * ) θ > θ ( f (ˆ x )- f ( x * )) θ = f (ˆ x )- f ( x * ) ≥ ....
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- Fall '08
- Optimization, Mathematical optimization, Fermat's theorem, FINITE DIMENSIONAL OPTIMIZATION