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# slide12-sept19 - ECON 401 Theorem of Khun and Tucker Siyang...

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ECON 401: Theorem of Khun and Tucker Siyang Xiong Rice University September 18, 2011 Xiong (Rice University) ECON 401 September 18, 2011 1 / 17

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Constrained optimization max f . x / s.t. g . x / D 0 and/or h . x / & 0 . Use the Theorem of Lagrange to solve max f . x / s.t. g . x / D 0 . Use the Theorem of Kuhn and Tucker to solve max f . x / s.t. h . x / & 0 . Xiong (Rice University) ECON 401 September 18, 2011 2 / 17
Constrained optimization max f . x / s.t. h . x / & 0 . I.e., max f . x / s.t. h 1 . x / & 0 ... h k . x / & 0 . with i D 1 , 2 , ..., k , we say h i . x / 0 is an e/ective constraint at x ± if h i . x ± / D 0. Xiong (Rice University) ECON 401 September 18, 2011 3 ± 17

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The Theorem of Lagrange Consider the constrained optimization problem: max f . x / s.t. g . x / D 0 . where f : R n ! R and g : R n ! R k are di/erentible functions. The Theorem of Lagrange: Suppose f achieves local maximum or minimum at x , and Rank . Dg . x // D k . Then, there exists D 1 , ..., k ± such that Df x ± C k X i D 1 i Dg i x ± D 0 Xiong (Rice University) ECON 401 September 18, 2011 4 ± 17
The Theorem of Kuhn and Tucker Consider the constrained optimization problem: max f . x / s.t. h . x / & 0 . where f : R n ! R and h : R n ! R k are di/erentible functions. The Kuhn-Tucker Theorem: Suppose f achieves local maximum or minimum at x ± . Let k denote the number of e/ective constraints at x ± . Further, suppose Rank . Dh . x ± // D k . Then, there exists ± D

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slide12-sept19 - ECON 401 Theorem of Khun and Tucker Siyang...

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