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slide9-sept12 - ECON 401: Theorem of Lagrange Siyang Xiong...

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Unformatted text preview: ECON 401: Theorem of Lagrange Siyang Xiong Rice University September 11, 2011 Xiong (Rice University) ECON 401 September 11, 2011 1 / 22 Constrained optimization max f . x / s.t. g . x / D 0, and/or h . x / & . Xiong (Rice University) ECON 401 September 11, 2011 2 / 22 Constrained optimization for example: max f . x / s.t. y D x 2 C p x can be written as max f . x / s.t. y & x 2 & p x D another example: max u . x / s.t. p x I , can be written as max u . x / s.t. I & p x . Xiong (Rice University) ECON 401 September 11, 2011 3 / 22 Constrained optimization: equality constraint Consider the constrained optimization problem: max f . x / s.t. g . x / D . 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 Xiong (Rice University) ECON 401 September 11, 2011 4 22 Example Consider the constrained optimization problem: max & & x 2 & y 2 s.t. x C y D 1 , i.e., f . x , y / D & & x 2 & y 2 and g . x , y / D x C y & 1. consider x D & 1 2 , 1 2 . Dg . x / D [ 1 , 1 ] , and Rank . Dg . x // D 1. Df . x / D [ & 2 x , & 2 y ] and hence Df . x / D [ & 1 , & 1 ] . we have Df & x C k X i D 1 & i Dg i & x D [ & 1 , & 1 ] C & 1 [ 1 , 1 ] let & 1 D 1. Then, Df & x C k X i D 1 & i Dg i & x D Xiong (Rice University) ECON 401 September 11, 2011 5 / 22 Example: necessity Vs su ciency Consider the constrained optimization problem: max & x 3 C y 3 s.t. x & y D , i.e., f . x , y / D x 3 C y 3 and g . x , y / D x & y ....
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This note was uploaded on 02/09/2012 for the course ECON 401 taught by Professor Siyang during the Spring '11 term at Rice.

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slide9-sept12 - ECON 401: Theorem of Lagrange Siyang Xiong...

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