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 xhix1 x2

xlo’ xhi’x2’x1’Second iteration

First iteration

618.0

)(

)(

2

1

lohilo

lohihi

xxxx

xxxx

5

Bounds on minimum

xl xu

x2’xl’ x1’Second iteration

First iteration

Fu

xu’

Fl

)(1' luuu xxxx

1.618(xu-xl)

6

Bounding minimum algorithmGiven, xl, Fl, xmax

Guess xu

Fu>Fl

Expandx1=xu*

xu=x1+1/(x1-xl)

Fu>F1

xl=x1

Minimum in [xl,xu]STOP

Y

N

Y

N Expand

* 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

y x1( ) 187.227 2.105 x1 0.053 x12

0 10 20 30 40165

170

175

180

F

y x1( )

x x1

8

Example of minimizing function using second degree polynomial approximation obtained through

regression. Five data points uniformly distributed between 10 and 30 are used

y x1( ) 174.962 0.845 x1 0.025 x12

0 10 20 30 40165

170

175

180

F

y x1( )

x x1

9

Example of minimizing function using second degree polynomial approximation obtained using

three data points (exact fit)

y x1( ) 175.665 0.903 x1 0.026 x12

0 10 20 30 40165

170

175

180

F

y x1( )

x x1

0.903

2 0.02617.365

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