how does microsoft excel solver deal with nonlinear problems – a simple example 劉亮志
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How does Microsoft Excel solver deal with nonlinear problems – A simple example
劉亮志
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Recall our question
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• The equation for GARCH(1,1) is
where
Set ,
Then we get
1
21
21
2 nnLn V
LV 0
21
21
2 nnn
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• Later, we met an example about MLE method :
Consider the problem of estimating the variance
of variable X from m observations on X when
underlying distribution is normal with zero mean.
Assume that the observations are .
Denote the variance v . The likelihood of being
observed is defined as the probability density function
for X when . This is
Then the joint density function of this m observations is
(**)
muuu ,...,, 21
iu
iuX
)2
exp(2
1 2
v
u
vi
)]2
exp(2
1[
2
1 v
u
vi
m
i
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Using MLE, the best estimate of v is the value that maximizes
the expression (**).
Maximizing (**) is same as maximizing
Let
Then we want to maximize the expression
])ln([2
1 v
uv i
m
i
21
21 nnv
m
i nn
inn u
uu
121
21
221
21 ]
)()ln([
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• We wonder what’s the value of that maximizes
subject to the boundary conditions
,,
m
i nn
inn u
uu
121
21
221
21 ]
)()ln([
1
0
1
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How does Microsoft Excel solver deal with nonlinear programs?
(1) the long …long way to find the method out
(2) basic idea and a simple example
(3) reference
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Long way
Microsoft Office Online
http://office.microsoft.com/zh-tw/
• Excel adopts the generalized reduced gradient (GCG)
method to deal with nonlinear problems.
• GCG originates from the method of reduced gradient
of Wolf.
• More specifically, the question (MLE) we met is a nonlinear
programming problem with linear constraints.
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Basic idea of method of reduced gradient
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Summary of Reduced Gradient Algorithm
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A simple example
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Iterative 1
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Iterative 2
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Iterative 3
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Reference
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• Mokhtar s. Bazraa, Hanif D. Sherali ,and C. M. Shetty. Nonlinear Programming: Theory and Algorithm second edition Section 10.6
• Wenyu Sun, and Ya-Xiang Yuan. Optimization Theory and Methods: Nonlinear Programming Section 11.3
• Hong-Tau Lee, Sheu-Hua Chen, He-Yau Kang.
A Study of Generalized Reduced Gradient Method
with Different Search Directions • Daniel Fylstra, Leon Lasdon, John Waston, Allen
Waren. Design and Use of the Microsoft Excel Solver.