slide12-sept19

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

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
ECON 401: Theorem of Khun and Tucker Siyang Xiong Rice University September 18, 2011 Xiong (Rice University) ECON 401 September 18, 2011 1 / 17
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 17

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online