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Numerical Methods Multidimensional Gradient Methods in Optimization- Example http://nm.mathforcollege.com. For more details on this topic Go to http://nm.mathforcollege.com Click on Keyword Click on Multidimensional Gradient Methods in Optimization. You are free. - PowerPoint PPT PresentationTRANSCRIPT
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Numerical Methods Multidimensional Gradient Methods in Optimization- Example
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For more details on this topic
Go to http://nm.mathforcollege.com Click on Keyword Click on Multidimensional Gradient
Methods in Optimization
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Example
Determine the minimum of the function
Use the poin (2, 1) as the initial estimate of the optimal solution.
5
42, 22 xyxyxf
)0(
)0()0(
y
xx
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SolutionIteration 1: To calculate the gradient; the partial derivatives must be evaluated asRecalled that
62)2(222
xx
f2)1(22
yy
f
j2i6 f
h
hhx
fhxx
i
ii
21
62
2
6
1
2)1(
)()1(
42),( 22 xyxyxf
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SolutionNow the function can be expressed along the direction of gradient as
To get ,we set
yxf ,
134040)( 2 hhhg
)(4)62(2)21()62()( 221 hghhhxf i
5.040800 hhdh
dgming
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Solution Cont.Iteration 1 continued: This is a simple function and it is easy to determine by taking the first derivative and solving for its roots.
This means that traveling a step size of along the gradient reaches a minimum value for the function in this direction. These values are substituted back to calculate a new value for x and y as follows:
50.0* h
5.0h
0)5.0(21
1)5.0(62
y
x
0.30,1 f 131,2 fNote that http://nm.mathforcollege.com
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Solution Cont.Iteration 2: The new initial point is .We calculate the gradient at this point as
0) ,1(
02)1(222
xx
f
0)0(22
yy
f
j)0(i)0(f
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Solution Cont.
This indicates that the current location is a local optimum along this gradient and no improvement can be gained by moving in any direction. The minimum of the function is at point (-1,0),and .
34)1(2)0()1( 22min f
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THE END
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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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