numerical methods newton’s method for one - dimensional optimization - theory
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Numerical Methods Newton’s Method for One -Dimensional Optimization - Theory
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Dimensional Optimization
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Newton’s Method-Overview
Open search method A good initial estimate of the solution is
required The objective function must be twice
differentiable Unlike Golden Section Search method
• Lower and upper search boundaries are not required (open vs. bracketing)
• May not converge to the optimal solution
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Newton’s Method-How it works The derivative of the
function ,Nonlinear root finding equation, at the function’s maximum and minimum.
The minima and the maxima can be found by applying the Newton-Raphson method to the derivative, essentially obtaining
Next slide will explain how to get/derive the above formula.
xfOpt. )(0' xFxf
)(
)(''
'
1i
iii xf
xfxx
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Slope @ pt. C ≈
We “wish” that in the next iteration xi+1 will be the root, or .
Thus:Slope @ pt. C =
F(x)
• •••
•
xixi+1
xi – xi+1
A
B
E F
C
D F(xi) - 0F(xi+1)
F(xi)
x
Or
Hence: N-R Equation
Newton’s Method-To find root of a nonlinear equation
1
1)()(
ii
ii
XX
XFXF
0)( 1 iXF
1
0)(
ii
i
XX
XF
1
)()(
ii
ii XX
XFXF
)(
)(1
I
iii XF
XFXX
)( iXFSlope
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If ,then . For Multi-variable case ,then N-R
method becomes
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Newton’s Method-To find root of a nonlineat equation
)()( xfxF )(
)(1
i
iii Xf
XfXX
)()]([ 11 iiii XfXfXX
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Newton’s Method-Algorithm
Initialization: Determine a reasonably good estimate for the maxima or the minima of the function .
Step 1. Determine and .Step 2. Substitute (initial estimate for the first
iteration) and into
to determine and the function value in iteration i.Step 3.If the value of the first derivative of the function is
zero then you have reached the optimum (maxima or minima). Otherwise, repeat Step 2 with the new value of
9
xf xf ' xf ''
ix 0x xf ' xf ''
)(
)(''
'
1i
iii xf
xfxx
1ix
ix
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Acknowledgement
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This material is based upon work supported by the National Science Foundation under Grant # 0717624. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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The End - Really
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Numerical Methods Newton’s Method for One -Dimensional Optimization - Example
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Dimensional Optimization
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work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).
Noncommercial — You may not use this work for commercial purposes.
Share Alike — If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one.
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18
Example
The cross-sectional area A of a gutter with equal base and edge length of 2 is given by
)cos1(sin4 A
.
Find the angle which maximizes the cross-sectional area of the gutter.
2
2
2
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Solution)cos1(sin4)( fThe function to be maximized is
Iteration 1: Use as the initial estimate of the solution
)sincos(cos4)( 22 f
)cos41(sin4)( f
rad7854.040
0466.1)
4cos41(
4sin4
)4
sin4
cos4
(cos4
4
22
1
196151.5)0466.1( f
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Solution Cont.Iteration 2:
0472.1)0466.1cos41(0466.1sin4
)0466.1sin0466.1cos0466.1(cos40466.1
22
2
Iteration
1 0.7854 2.8284 -10.8284 1.0466 5.1962
2 1.0466 0.0062 -10.3959 1.0472 5.1962
3 1.0472 1.06E-06 -10.3923 1.0472 5.1962
4 1.0472 3.06E-14 -10.3923 1.0472 5.1962
5 1.0472 1.3322E-15 -10.3923 1.0472 5.1962
Summary of iterations
Remember that the actual solution to the problem is at 60 degrees or 1.0472 radians.
)(festimate)(' f )('' f
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Acknowledgement
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This material is based upon work supported by the National Science Foundation under Grant # 0717624. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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The End - Really