1. L23 Numerical Methods part 3
Project
Homework
Review
Steepest Descent Algorithm
Summary
Test 4 results

2. H22 ans optimum solution __0.444___ min value __0.0494__
interval of uncertainty__0.889__
number of fcn evals __6____

3. H22 cont’d
For iterations # 2 on….The interval is reduced to 0.888/1.333 =67% …. For the cost of 2 function evaluations. If we create a measure of efficiency

4. H22

5. H22 optimum solution __.472____ min value __.0124______
interval of uncertainty___0.764_____
number of fcn evals ___5_____
For iterations # 2 on….The interval is reduced to 61.8% of interval, I for the cost of only 1 function evaluation. If we create a measure of efficiency
Golden
Section
Best

6. Search algorithm? 1. Find a direction, then
2. Find best step size for alpha
3. Repeat steps 1 and 2 ‘til “done”

7. Unimodal functions in “locale”
monotonic decreasing then monotonic increasing
monotonic increasing then monotonic decreasing
Figure 10.4 Unimodal function f().

8. Review: Step Size Methods
“Analytical”
Search direction = (-) gradient, (i.e. line search)
Find f’(α)=0, f’’(α)≥0
Region Elimination (“interval reducing”)
Equal interval
Alternate equal interval
Golden Section
Others
Newton-Raphson
Successive quadratic Interpolation

9. Successive Alternate Equal Interval
Assume bounding phase has found
Min can be on either side of
Point values… not a line
But for sure its not in this region!

10. Golden section
Figure 10.9 Graphic of a
section partition.

11. Descent Algorithm?
Descent is guaranteed!

12. Steepest descent algorithm
How does it work?

13. “Modified” Steepest-Descent Algorithm

14. Ex 10.4
Use Solver to find α*

15. EX 10.4
||c||=0
Done!

16. H22 Prob 10.52
Let’s use SteepDescentTemplate.xls to set up and solve.

17. Summary Step size methods: analytical, region elimin.
Golden Section is very efficient
Algorithms include stopping criteria (||c||,∆f )
Steepest descent algorithm
Convergence is assured
Lots of Fcn evals (in line search)
Each iteration is independent of previous moves (i.e. totally “local” )
Successive iterations slow down.. may stall