Lecture 6

# Then 0 12 ispositivedefinite a if

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Unformatted text preview: mean for the Hessian to be negative definite. We have: be the ith‐order Theorem: Let ∶ → be twice continuously differentiable, and let principal matrix of the Hessian . Then 0, 1,2, … , is positive definite. A. If alternates in sign, starting with 0, is negative definite. B. If is not negative definite that the Note, as a matter of logic, this does not mean that if function f is not concave. All it is saying is that if we can establish the that a function’s Hessian is negative definite, the function is concave. Bottom Line: This theorem, coupled with our earlier theorem connecting negative and positive definiteness to the concavity or convexity of the function, respectively, informs us ∗ 0. whether the function attains a local max or a min at a point where the gradient Summary 1. Concave and quasiconcave. A real valued function that is concave is quasiconcave (The converse is not necessarily true). 2. Quasiconcave and superior sets. A real value function is quasiconcave iff its superior sets are convex. 3. Second‐order conditions for an optima. A real valued function : ⟶ reaches a local ∗ interior maximum if its Hessian is negative semidefinite. Here, ∗ is a critical point of . Likewise, A real valued function : ⟶ reaches a local interior minimum if its Hessian ∗ is positive semidefinite. ∗ 4. Sufficient conditions for negative semidefiniteness. Let be the ith principal minor of ∗ ∗ ∗ ∗ . If the alternates in sign, starting with the first, 0, then is ∗ negative semidefinite. If 0 for all i, i = 1, ..., K, then the Hessian is positive semidefinite. ∗ ∗ 0 for i = 1, ..., K and 5. Sufficient conditions for a local interior optima. If ∗ alternates in sign, starting with the first, 0, reaches a local max at ∗ .(Replace ∗ condition on the determinant of the principal minors to 0 for all i, reaches a local min at ∗ ). 6. Critical points of a strictly concave (strictly convex) function. A critical point of a strictly concave (convex) function is a unique global maximum (minimum). Look...
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