Quadratic Forms and Objective functions with two or more variables

1. Quadratic Forms and Objective functions with two or more variables

2. Two Choice Variables Quadratic Forms • Let • Then ------------- • can be written as ----------------- Definiteness • Positive definite if q is invariably positive (q > 0) • Positive semidefiniteif q is invariably nonnegative () • Negative definite if q is invariably negative (q < 0) • Negative semidefiniteif q is invariably non positive ()

3. Rewriting the quadratic form using completing square • Writing in Matrix Form We have The determinant of the coefficient matrix is important in determining the sign of

4. Writing equation (2) in terms of equation (1) gives further insights • Since • Conditions for

5. Discriminant of a Quadratic form • In general the discriminant of a quadratic form Is the symmetric determinant • In the particular case of quadratic form The discriminant is the determinant with second order partial derivatives as its elements.

6. Hessian Determinant Determinant with all the second order partial derivatives is called Hessian Matrix.

7. Extremum Conditions for two choice variable So,

8. Objective Functions with 3 choice Variables • First order condition: • Second order condition :

9. Hessian Matrix for 3 choice Variables Optimization Conditions First Order Necessary condition: Second order Conditions

10. Variable Case • Optimization Conditions First order Necessary conditions Second order Conditions