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

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ECON 401: Theorem of Lagrange Siyang Xiong Rice University September 11, 2011 Xiong (Rice University) ECON 401 September 11, 2011 1 / 22

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Constrained optimization max f . x / s.t. g . x / D 0, and/or h . x / ° 0 . 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 0 another example: max u . x / s.t. p ² x ³ I , can be written as max u . x / s.t. I ± p ² x ° 0 . Xiong (Rice University) ECON 401 September 11, 2011 3 / 22

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Constrained optimization: equality constraint 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 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 µ [ 1 , 1 ] let ° ´ 1 D 1. Then, Df ° x ´ ± C k X i D 1 ° ´ i Dg i ° x ´ ± D 0 Xiong (Rice University) ECON 401 September 11, 2011 5 / 22

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Example: necessity Vs su¢ ciency Consider the constrained optimization problem: max ° x 3 C y 3 ± s.t. x ± y D 0 , i.e., f . x , y / D x 3 C y 3 and g . x , y / D x ± y . consider x ´ D . 0 , 0 / . Dg . x / D [ 1 , ± 1 ] , and Rank . Dg . x ´ // D 1. Df . x / D ² 2 x 2 , 2 y 2 ³ and hence Df . x ´ / D [ 0 , 0 ] . consider ° ´ D 0, we have Df ° x ´ ± C k X i D 1 ° ´ i Dg i ° x ´ ± D [ 0 , 0 ] C 0 µ [ 1 , ± 1 ] D 0 however, . 0 , 0 / is neither a local minimum nor a local maximum: for small " > 0, we have f . ± " , ± "/ D ± 2 " 3 < 0 D f . 0 , 0 / D 0 < 2 " 3 D f ." , "/ Xiong
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slide9-sept12 - ECON 401 Theorem of Lagrange Siyang Xiong...

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