1. Optimization of functions of one variable (Section 2)
Find minimum of function of one variable
Occurs directly
Part of iterative algorithm (line search)
Unimodal function, single optimum -- step toward optimum results in reduction of objective function along the path

2. Two methods Golden section search Polynomial approximation
Golden section search; known convergence rate, guaranteed to find interval bounding optimum (tolerance interval). Provides information about confidence in solution. Expensive
Polynomial approximation. Efficient but not as robust as Golden section search

3. Golden section search
Starts with interval known to contain minimum (tolerance interval)
Proceeds by narrowing tolerance interval
Uses four data points for which objective function is evaluated.
In each iteration -- one additional function evaluation
Tolerance interval reduces to 61.8% of interval from previous iteration

4. Golden section method xlo xhi x1 x2 xlo’ xhi’ x2’ x1’ Second iteration
First iteration

5. Bounds on minimum Fl Fu 1.618(xu-xl) xu First iteration xl
Second iteration
xl’
x1’
x2’
xu’

6. Bounding minimum algorithm
Given, xl, Fl, xmax
Guess xu Y
Fu>Fl
Minimum in [xl,xu]
STOP N
Expand
x1=xu*
xu=x1+1/(x1-xl) Y
Fu>F1 N
Expand
xl=x1
* Stop if xu>xmax

7. Example of minimizing function using second degree polynomial approximation obtained through regression. Four data points are used from minimum bounding solution

8. Example of minimizing function using second degree polynomial approximation obtained through regression. Five data points uniformly distributed between 10 and 30 are used

9. Example of minimizing function using second degree polynomial approximation obtained using three data points (exact fit)

10. Minimizing constrained functions of one variable
Direct approach
Deal with each function (objective, constraint) individually
Indirect approach
Develop and use pseudo objective function that includes both the objective function and the constraint functions