1. Optimization Part II
G.Anuradha

2. Review of previous lecture- Steepest Descent
Choose the next step so that the function decreases:
For small changes in x we can approximate F(x):
where
If we want the function to decrease:
We can maximize the decrease by choosing:

3. Example

4. Plot

7. Necessary and sufficient conditions for a function with single variable

9. Functions with two variables
Sufficient conditions
Necessary conditions

10. Stationary Points

11. Effect of learning rate
More the learning rate the trajectory becomes oscillatory.
This will make the algorithm unstable
The upper limit for learning rates can be set for quadratic functions

12. Stable Learning Rates (Quadratic)
Stability is determined
by the eigenvalues of
this matrix.
Eigenvalues
of [I - aA].
(li - eigenvalue of A)
Stability Requirement:

13. Example

14. Newton’s Method Take the gradient of this second-order approximation
and set it equal to zero to find the stationary point:

15. Example

16. Plot

17. This is used for finding line minimization methods and their stopping criteria
Initial bracketing
Line searches
Newton’s method
Secant method
Sectioning method

18. Initial Bracketing
Helps in finding the range which contains the relative minimum
Bracketing some assumed minimum in the starting interval is required
Two schemes are used for this purpose

20. Sectioning methods