1. L24 Numerical Methods part 4
Project Quesitons
Project problem added
3 position linkage synthesis
See Norton(2nd Ed) Chp 5
Homework
Review
Conjugate Gradient Algorithm
Summary
Excel – User defined function

2. H22

3. 10.52

4. 10.57

5. 10.61

6. Fractional Reduction Alternate Equal Interval Golden Section
aka “Exhaustive”
Add these formulas to your notes for next test!

7. Steepest descent algorithm
How does it work?

8. “Modified” Steepest-Descent Algorithm

9. Cauchy’s Method
Engineering Optimization
Ravindran, Ragsdell,Reklaitus

10. Algorithms Algorithms include stopping criteria (||c||,∆f )
Steepest descent algorithm
Convergence is assured
However, lots of Fcn evals (in line search)
Each iteration is independent of previous moves (i.e. totally “local” )
Successive iterations slow down.. may stall
Need non-stalling algorithm?

11. Conjugate Gradient

12. Conj grad ex

13. Conj grad ex contd

14. “Deflected” Steepest Descent
A comparison of the convergence of gradient descent with optimal step size (in green) and conjugate vector (in red) for minimizing a quadratic function associated with a given linear system. Conjugate gradient, assuming exact arithmetic, converges in at most n steps where n is the size of the matrix of the system (here n=2). Wik

15. Higher Order Methods

16. Summary Golden Section is very efficient
Algorithms include stopping criteria (||c||,∆f )
Steepest descent algorithm may stall
Conjugate Gradient
Convergence in n iterations (n=# of design var’s)
Lots of Fcn evals (in line search)
May need to restart after n+1 iterations