Definitions and Some Theorems

Interiorminimumifitshessian be the ith 4 sufficient

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Unformatted text preview: traints is less than the number of variables). Theorem: (Arrow and Enthoven) Let ∶ → 1. Concave and quasiconcave. A real valued function that is concave is quasiconcave (The converse is not necessarily true). . Let ∗ 0 ⋮ ⋯ ⋱ ⋯ ⋮ , i.e., ⋯ ∗ ⋱ ⋮ bordered by the row is the matrix of second derivatives, ⋮ ⋯ and column vectors of first derivatives and the element 0. Let 0 0 ⋯ 0 ⋮ ⋱ ⋮ , ,…, . … , 3. Second‐order conditions for an optima. A real valued function : ⟶ ∗ is negative semidefinite. Here, reaches a local interior maximum if its Hessian ∗ is a critical point of . Likewise, A real valued function : ⟶ reaches a local ∗ is positive semidefinite. interior minimum if its Hessian ∗ be the ith 4. Sufficient conditions for negative semidefiniteness. Let ∗ ∗ . If the alternates in sign, starting with the first, principal minor of ∗ ∗ ∗ 0, then is negative semidefinite. If 0 for all i, i = 1, ..., K, then the Hessian is positive semidefinite. ∗ 5. Sufficient conditions for a local interior optima. If 0 for i = 1, ..., K and ∗ ∗ alternates in sign, starting with the first, 0, reaches a...
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