**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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