Question

y=fx1, ……..,x5)

St:

c1=w1(x1, ……..,x5)

c2=w2(x1, ……..,x5)

By using the signs of the principal minors Hj

1-Derive the second- order-sufficient condition for maximum.

2-Derive the second order-sufficient condition for minimum

Answer 1

Solution

(1)

Given

on the constraint set

As usual we consider the Lagrangian

and the following bordered Hessian matrix

This matrix has leading principal minors

The first two matrices are zero matrices.

Next matrix    have zero determinant.

The determinant of the next minor is where is the upper minor of after block of zeros, so does not contain information about .

And only the determinants of last 3 leading principal minors

carry information about both, the objective function and the constraints .

Suppose that satisfies the conditions

a)  

b) There exists   such that is a critical point of .

c) For the bordered Hessian the last 3 leading principal minors

evaluated at alternate in sign where the last minor has the sign as

Then is a local max in .

(2)

Take minors of the Hessian H from (1)

Then

Suppose that satisfies the conditions

a)  

b) There exists   such that is a critical point of .

c) For the bordered Hessian all the last 3 leading principal minors

evaluated at have the same sign as ,

Then is a local min in .

Conclution

This table describes the above sign patterns:

			...
max			...
min			...

where k=number of constraints equations , n=number of variables