n16 - Atsushi Inoue ECG 765 Mathematical Methods for...

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Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2008 Lecture Notes 16 Optimization Subject to Mixed Constraints We will consider cases in which there are equality constraints as well as inequality constraints in maximization and minimization problems. Theorem (Necessary Conditions for Maximization with Mixed Constraints). Sup- pose that f , g 1 ,..., g k , h 1 ,..., h m are C 1 functions. Suppose that x * is a local maximizer of f on the constraint set defined by the k inequalities and m equalities: g 1 ( x ) b 1 , ..., g k ( x ) b k h 1 ( x ) = c 1 , ..., h k ( x ) = c m . Suppose that the following constraint qualification is satisfied: the rank of the Jacobian matrix of the equality constraints and the binding inequality constraints is k 0 + m where k 0 is the number of binding inequality constraints. Form the Lagrangian L ( x,λ,μ ) = f ( x ) + k X i =1 λ i [ b i - g i ( x )] + m X i =1 μ j [ c j - h j ( x )] . Then there exists multipliers
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n16 - Atsushi Inoue ECG 765 Mathematical Methods for...

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