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n14 - Atsushi Inoue ECG 765 Mathematical Methods for...

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Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 14 Second-Order Conditions for Equality-Constrained Optimization Because the Lagrange theorem provides first-order necessary conditions for constrained local optima only, the solution to the first-order conditions can be a constrained local maximum, constrained local minimum or neither. We will characterize second-order sufficient conditions for constraint local optima. Theorem. Consider max f ( x ) subject to g ( x ) = c Suppose that (a) x * satisfies g ( x * ) = c , (b) There exists λ * such that ( x * , λ * ) is a critical point of the Lagrangian, (c) The Hessian of the Lagrangian with respect to x at ( x * , λ * ), 2 x L ( x * , λ * ), is negative definite on the linear constraint set { v : Dg ( x * ) v = 0 } ; that is, v 6 = 0 and Dg ( x * ) v = 0 v > 2 x L ( x * , λ * ) v < 0 . Then x * is a strict local constrained max. Theorem . Consider min f ( x ) subject to g ( x ) = c Suppose that (a) x * satisfies g ( x * ) = c , (b) There exists λ * such that ( x * , λ * ) is a critical point of the Lagrangian,
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