1. Introduction to Optimization Anjela Govan North Carolina State University SAMSI NDHS Undergraduate workshop 2006

2. What is Optimization? Optimization is the mathematical discipline which is concerned with finding the maxima and minima of functions, possibly subject to constraints.

3. Where would we use optimization? Architecture Nutrition Electrical circuits Economics Transportation etc.

4. What do we optimize? A real function of n variables with or without constrains

5. Unconstrained optimization

6. Optimization with constraints

7. Lets Optimize Suppose we want to find the minimum of the function

8. Review max-min for R 2 What is special about a local max or a local min of a function f (x)? at local max or local min f ’(x)=0 f ”(x) > 0 if local min f ”(x) < 0 if local max

9. Review max-min for R 3

10. Second Derivative Test Local min, local max, saddle point Gradient of f – vector (d f/ dx, d f /dy, d f /dz ) direction of fastest increase of f Global min/max vs. local min/max

11. Gradient Descent Method Examples Minimize function

12. Gradient Descent Method Examples Use function gd(alpha,x0)  Does gd.m converge to a local min? Is there a difference if  > 0 vs.  < 0?  How many iterations does it take to converge to a local min? How do starting points x0 affect number of iterations? Use function gd2(x0)  Does gd2.m converge to a local min?  How do starting points x0 affect number of iterations and the location of a local minimum?

13. How good are the optimization methods? Starting point Convergence to global min/max. Classes of nice optimization problems Example: f(x,y) = 0.5(  x 2 +y 2 ),  > 0 Every local min is global min.

14. Other optimization methods Non smooth, non differentiable surfaces  can not compute the gradient of f  can not use Gradient Method Nelder-Mead Method Others

15. Convex Hull A set C is convex if every point on the line segment connecting x and y is in C. The convex hull for a set of points X is the minimal convex set containing X.

16. Simplex A simplex or n-simplex is the convex hull of a set of (n +1). A simplex is an n- dimensional analogue of a triangle. Example:  a 1-simplex is a line segment  a 2-simplex is a triangle  a 3-simplex is a tetrahedron  a 4-simplex is a pentatope

17. Nelder-Mead Method n = number of variables, n+1 points form simplex using these points; convex hull move in direction away from the worst of these points: reflect, expand, contract, shrink Example:  2 variables  3 points  simplex is triangle  3 variables  4 points  simplex is tetrahedron

18. Nelder-Mead Method – reflect, expand

19. Nelder-Mead Method-reflect, contract

20. A tour of Matlab: Snapshots from the minimization After 0 steps

21. A tour of Matlab: Snapshots from the minimization After 1 steps

22. A tour of Matlab: Snapshots from the minimization After 2 steps

23. A tour of Matlab: Snapshots from the minimization After 3 steps

24. A tour of Matlab: Snapshots from the minimization After 7 steps

25. A tour of Matlab: Snapshots from the minimization After 12 steps

26. A tour of Matlab: Snapshots from the minimization After 30 steps (converged)

27. fminsearch function parameters: q =[C,K] cost function: Minimize cost function [q,cost]= fninsearch(@cost_beam, q0,[],time,y_tilde)

28. Our optimization problem In our problem  Our function:  cost function “lives” in R 3  2 parameters C and K, n=2  Simplex is a triangle

29. Done!