6.Bordered Hessian matrix


Bordered Hessian Matrix

 Hessian matrix of    :

   H(f) = \begin{bmatrix} \dfrac{\partial^2 f}{\partial x_1^2} & \dfrac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_1\,\partial x_n} \\[2.2ex] \dfrac{\partial^2 f}{\partial x_2\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_2^2} & \cdots & \dfrac{\partial^2 f}{\partial x_2\,\partial x_n} \\[2.2ex] \vdots & \vdots & \ddots & \vdots \\[2.2ex] \dfrac{\partial^2 f}{\partial x_n\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_n^2} \end{bmatrix}.

 Bordered Hessian Matrix(경계 헤시안 행렬) of  the Lagrange- function(2013.12.02)   : (0.Calculus 참고 )

    :

   

       

 If the constraints are linear(제한함수가 선형일 경우) -- as    the bordered Hessian becomes :

   

 In particular, given the  function as before: f(x_1, x_2, \dots, x_n), adding a constraint (linear)(하나의 선형제한함수일 경우) function such that: g(x_1, x_2, \dots, x_n) = c,   the bordered Hessian appears as (참고: http://en.wikipedia.org/wiki/Hessian_matrix 2013.12.03)

 H(f,g) = \begin{bmatrix} 0 & \dfrac{\partial g}{\partial x_1} & \dfrac{\partial g}{\partial x_2} & \cdots & \dfrac{\partial g}{\partial x_n} \\[2.2ex] \dfrac{\partial g}{\partial x_1} & \dfrac{\partial^2 f}{\partial x_1^2} & \dfrac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_1\,\partial x_n} \\[2.2ex] \dfrac{\partial g}{\partial x_2} & \dfrac{\partial^2 f}{\partial x_2\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_2^2} & \cdots & \dfrac{\partial^2 f}{\partial x_2\,\partial x_n} \\[2.2ex] \vdots & \vdots & \vdots & \ddots & \vdots \\[2.2ex] \dfrac{\partial g}{\partial x_n} & \dfrac{\partial^2 f}{\partial x_n\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_n^2} \end{bmatrix}

Theorem6 (Bordered Hessian Test)

 For the bordered Hessian matrix of lagrage function with  and m constraints  :

1) The Hessian matrix of   at critical points is negative definite (i. e.  has local maximum(극대)) under the constraints 

   iff its Leading principal minor of bordered Hessian alternative in signs and  

2) The Hessian matrix of   at critical points is positive definite (i. e.  has local minimum(극소)) under the constraints 

   iff its Leading principal minor of bordered Hessian are all negative and also  .

 

The proof of bordered Hessian Test : artaci, http://goo.gl/IIkp5m 2013.12.02
Information of bordered Hessian Matrix : Ronald wendner, http://goo.gl/PK6Fgu 2013. 12.02
Application Example of Bordered Hessian matrix : Tianxi Wang, http://goo.gl/E8OeIJ 2013.12.02

Example) Find the local maximum and local minimum points of 

subject to the constraint  . (The same example of HOME application)

Sol)

Find the critical points of   under the constraint   using the Lagrange multiplier.

Then,  .




Then, the critical points are  .




The Bordered Hessian matrix of  under  is   .




At   ,    and   

Thus  has local maximum (i.e. negative definite) at  by Bordered Hessian Test.




At   ,    and   

Thus  has local maximum (i.e. negative definite) at  by Bordered Hessian Test.




At   ,   and   

Thus  has local minimum (i.e. positive definite) at  by Bordered Hessian Test.




http://goo.gl/vxcrD3 2013.12.03

Example) 

Find the local maximum and local minimum point of

  

subject to the constraint   .




Example) 

Find the local maximum and local minimum point of

       

subject to the constraint   .




일반적인 경우)

http://goo.gl/74YCl8 .2013.12.03

다음은  제한하는 함수가 2개 있을 경우에 이변수함수의 극소값과 극대값을 판단해주는 interact입니다.




다음은 제한함수가 2개 있을 경우 삼변수 함수의 극대값과 극소값을 판단해주는 interact입니다.




Back TO the HOME