Optimization of functions of one variable (Section 2)
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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
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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
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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
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Golden section method
xlo xhix1 x2
xlo’ xhi’x2’x1’Second iteration
First iteration
618.0
)(
)(
2
1
lohilo
lohihi
xxxx
xxxx
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Bounds on minimum
xl xu
x2’xl’ x1’Second iteration
First iteration
Fu
xu’
Fl
)(1' luuu xxxx
1.618(xu-xl)
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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
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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
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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
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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
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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