- 107 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Optimization with Equality Constraints' - prescott-houston

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

Extreme Values of a Function of two Variables

Saddle point

First Order Conditions(The necessary conditions)

Given the problem of maximizing ( or minimizing) of the objective function:

Z=f(x ,y )

Finding the Stationary Values solutions of the following system:

Examples

- Z=f(x,y)=x2+y2
- Z=f(x,y)=x2-y2
- Z=f(x,y)=xy

Second Order Conditions

The Hessian Matrix

H(x0,y0)>0 fxx >0 minimum

H(x0,y0)>0 fxx <0 maximum

H(x0,y0)<0 saddle

Extreme Values of a Function of two Variables

The method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function:

subject to constraints:

Extreme Values of a Function of two Variables

For instance minimize the objective function

Subject to the constraint:

Lagrangemultiplier

We introduce a new variable (λ) called a Lagrange multiplier, and study the Lagrange function:

Second Order Conditions

Bordered Hessian Matrix of the Second Order derivative is given by

The point is a minimum

The point is a maximum

Lagrangemultiplier

Given the problem of maximizing ( or minimizing) of the objective function

with constraints

First OrderConditionsNecessaryCondition

We build a Lagrangian function :

Finding the Stationary Values:

Second Order ConditionsSufficient Conditions

- Second order conditions:
- We must check the sign of a Bordered Hessian:

n=2 e m=1 the Bordered Hessian Matrix of the Second Order derivative is given by

Det>0 imply Maximum

Det<0 imply Minimum

Download Presentation

Connecting to Server..