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\begin_layout Title
Mini-HOWTO on using Octave for Unconstrained Nonlinear Optimization
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
Author : Etienne Grossmann
\family typewriter
<etienne@egdn.net>
\family default
(soon replaced by
\begin_inset Quotes eld
\end_inset
Octave-Forge developers
\begin_inset Quotes erd
\end_inset
?).
This document is free documentation; you can redistribute it and/or modify
it under the terms of the GNU Free Documentation License as published by
the Free Software Foundation.
\begin_inset Newline newline
\end_inset
.
\begin_inset space ~
\end_inset
\begin_inset space ~
\end_inset
\begin_inset space ~
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This is distributed in the hope that it will be useful, but WITHOUT ANY
WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE.
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\begin_layout Standard
\begin_inset Note Comment
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\begin_layout Plain Layout
Keywords: nonlinear optimization, octave, tutorial, Nelder-Mead, Conjugate
Gradient, Levenberg-Marquardt
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\emph on
This document refers to the frontend function
\begin_inset Quotes eld
\end_inset
minimize
\begin_inset Quotes erd
\end_inset
, which is now deprecated and will be removed in a future version of the
optim package.
An alternative frontend function is
\begin_inset Quotes eld
\end_inset
nonlin_min
\begin_inset Quotes erd
\end_inset
, not described here.
The backends of
\begin_inset Quotes eld
\end_inset
minimize
\begin_inset Quotes erd
\end_inset
(
\begin_inset Quotes eld
\end_inset
nelder_mead_min
\begin_inset Quotes erd
\end_inset
,
\begin_inset Quotes eld
\end_inset
bfgsmin
\begin_inset Quotes erd
\end_inset
,
\begin_inset Quotes eld
\end_inset
d2_min
\begin_inset Quotes erd
\end_inset
) can also be called directly.
This document will be removed in the future.
\end_layout
\begin_layout Standard
Nonlinear optimization problems are very common and when a solution cannot
be found analytically, one usually tries to find it numerically.
This document shows how to perform unconstrained nonlinear minimization
using the Octave language for numerical computation.
We assume to be so lucky as to have an initial guess from which to start
an iterative method, and so impatient as to avoid as much as possible going
into the details of the algorithm.
In the following examples, we consider multivariable problems, but the
single variable case is solved in exactly the same way.
\end_layout
\begin_layout Standard
All the algorithms used below return numerical approximations of
\emph on
local minima
\emph default
of the optimized function.
In the following examples, we minimize a function with a single minimum
(Figure
\begin_inset space ~
\end_inset
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:function"
\end_inset
), which is relatively easily found.
In practice, success of optimization algorithms greatly depend on the optimized
function and on the starting point.
\end_layout
\begin_layout Section*
\end_layout
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selectfont A simple example
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\begin_inset Caption
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\begin_inset CommandInset label
LatexCommand label
name "fig:function"
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2D slice of the function that is minimized throughout this tutorial.
Although not obvious at first sight, it has a unique minimum.
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\begin_layout Standard
We will use a call of the type
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[x_best, best_value, niter] = minimize (func, x_init)
\end_layout
\begin_layout Standard
to find the minimum of
\begin_inset Formula
\[
\begin{array}{cccc}
f\,: & \left(x_{1},.x_{2},x_{3}\right)\in\R^{3} & \longrightarrow & \left(x_{1}-1\right)^{2}/9+\left(x_{3}-1\right)^{2}/9+\left(x_{3}-1\right)^{2}/9\\
& & & -\cos\left(x_{1}-1\right)-\cos\left(x_{2}-1\right)-\cos\left(x_{3}-1\right).
\end{array}
\]
\end_inset
\end_layout
\begin_layout Standard
The following commands should find a local minimum of
\begin_inset Formula \ensuremath{f()}
\end_inset
, using the Nelder-Mead (aka
\begin_inset Quotes eld
\end_inset
downhill simplex
\begin_inset Quotes erd
\end_inset
) algorithm and starting from a randomly chosen point
\family typewriter
x0
\family default
\begin_inset space ~
\end_inset
:
\end_layout
\begin_layout LyX-Code
function cost = foo (xx)
\end_layout
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xx--;
\end_layout
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cost = sum (-cos(xx)+xx.^2/9);
\end_layout
\begin_layout LyX-Code
endfunction
\end_layout
\begin_layout LyX-Code
x0 = [-1, 3, -2];
\end_layout
\begin_layout LyX-Code
[x,v,n] = minimize ("foo", x0)
\end_layout
\begin_layout Standard
The output should look like
\begin_inset space ~
\end_inset
:
\end_layout
\begin_layout LyX-Code
x =
\end_layout
\begin_layout LyX-Code
1.00000 1.00000 1.00000
\end_layout
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\end_layout
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v = -3.0000
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n = 248
\end_layout
\begin_layout Standard
This means that a minimum has been found in
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\end_inset
and that the value at that point is
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\end_inset
.
This is correct, since all the points of the form
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\end_inset
, for some
\begin_inset Formula \ensuremath{i,j,k\in\N}
\end_inset
, minimize
\begin_inset Formula \ensuremath{f()}
\end_inset
.
The number of function evaluations, 248, is also returned.
Note that this number depends on the starting point.
You will most likely obtain different numbers if you change
\family typewriter
x0
\family default
.
\end_layout
\begin_layout Standard
The Nelder-Mead algorithm is quite robust, but unfortunately it is not very
efficient.
For high-dimensional problems, its execution time may become prohibitive.
\end_layout
\begin_layout Section*
\end_layout
\begin_layout Standard
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\backslash
fontfamily{cmss}
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selectfont Using the first differential
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Fortunately, when a function, like
\begin_inset Formula \ensuremath{f()}
\end_inset
above, is differentiable, more efficient optimization algorithms can be
used.
If
\family typewriter
minimize()
\family default
is given the differential of the optimized function, using the
\family typewriter
"df"
\family default
option, it will use a conjugate gradient method.
\end_layout
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## Function returning partial derivatives
\end_layout
\begin_layout LyX-Code
function dc = diffoo (x)
\end_layout
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x = x(:)' - 1;
\end_layout
\begin_layout LyX-Code
dc = sin (x) + 2*x/9;
\end_layout
\begin_layout LyX-Code
endfunction
\end_layout
\begin_layout LyX-Code
[x, v, n] = minimize ("foo", x0, "df", "diffoo")
\end_layout
\begin_layout Standard
This produces the output
\begin_inset space ~
\end_inset
:
\end_layout
\begin_layout LyX-Code
x =
\end_layout
\begin_layout LyX-Code
1.00000 1.00000 1.00000
\end_layout
\begin_layout LyX-Code
v = -3
\end_layout
\begin_layout LyX-Code
n =
\end_layout
\begin_layout LyX-Code
108 6
\end_layout
\begin_layout Standard
The same minimum has been found, but only 108 function evaluations were
needed, together with 6 evaluations of the differential.
Here,
\family typewriter
diffoo()
\family default
takes the same argument as
\family typewriter
foo()
\family default
and returns the partial derivatives of
\begin_inset Formula \ensuremath{f()}
\end_inset
with respect to the corresponding variables.
It doesn't matter if it returns a row or column vector or a matrix, as
long as the
\begin_inset Formula \ensuremath{i\nth}
\end_inset
element of
\family typewriter
diffoo(x)
\family default
is the partial derivative of
\begin_inset Formula \ensuremath{f()}
\end_inset
with respect to
\begin_inset Formula \ensuremath{x_{i}}
\end_inset
.
\end_layout
\begin_layout Section*
\end_layout
\begin_layout Standard
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
fontfamily{cmss}
\backslash
selectfont Using numerical approximations of the first differential
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Sometimes, the minimized function is differentiable, but actually writing
down its differential is more work than one would like.
Numerical differentiation offers a solution which is less efficient in
terms of computation cost, but easy to implement.
The
\family typewriter
"ndiff"
\family default
option of
\family typewriter
minimize()
\family default
uses numerical differentiation to execute exactly the same algorithm as
in the previous example.
However, because numerical approximation of the differentia is used, the
outpud may differ slightly
\begin_inset space ~
\end_inset
:
\end_layout
\begin_layout LyX-Code
[x, v, n] = minimize ("foo", x0, "ndiff")
\end_layout
\begin_layout Standard
wich yields
\begin_inset space ~
\end_inset
:
\end_layout
\begin_layout LyX-Code
x =
\end_layout
\begin_layout LyX-Code
1.00000 1.00000 1.00000
\end_layout
\begin_layout LyX-Code
v = -3
\end_layout
\begin_layout LyX-Code
n =
\end_layout
\begin_layout LyX-Code
78 6
\end_layout
\begin_layout Standard
Note that each time the differential is numerically approximated,
\family typewriter
foo()
\family default
is called 6 times (twice per input element), so that
\family typewriter
foo()
\family default
is evaluated a total of (78+6*6=) 114 times in this example.
\end_layout
\begin_layout Section*
\end_layout
\begin_layout Standard
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
fontfamily{cmss}
\backslash
selectfont Using the first and second differentials
\end_layout
\end_inset
\end_layout
\begin_layout Standard
When the function is twice differentiable and one knows how to compute its
first and second differentials, still more efficient algorithms can be
used (in our case, a variant of Levenberg-Marquardt).
The option
\family typewriter
"d2f"
\family default
allows to specify a function that returns the value of the function, the
first and second differentials of the minimized function.
Entering the commands
\begin_inset space ~
\end_inset
:
\end_layout
\begin_layout LyX-Code
function [c, dc, d2c] = d2foo (x)
\end_layout
\begin_layout LyX-Code
c = foo(x);
\end_layout
\begin_layout LyX-Code
dc = diffoo(x);
\end_layout
\begin_layout LyX-Code
d2c = diag (cos (x(:)-1) + 2/9);
\end_layout
\begin_layout LyX-Code
end
\end_layout
\begin_layout LyX-Code
[x,v,n] = minimize ("foo", x0, "d2f", "d2foo")
\end_layout
\begin_layout Standard
produces the output
\begin_inset space ~
\end_inset
:
\end_layout
\begin_layout LyX-Code
x =
\end_layout
\begin_layout LyX-Code
1.0000 1.0000 1.0000
\end_layout
\begin_layout LyX-Code
v = -3
\end_layout
\begin_layout LyX-Code
n =
\end_layout
\begin_layout LyX-Code
34 5
\end_layout
\begin_layout Standard
This time, 34 function evaluations, and 5 evaluations of
\family typewriter
d2foo()
\family default
were needed.
\end_layout
\begin_layout Section*
\end_layout
\begin_layout Standard
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
fontfamily{cmss}
\backslash
selectfont Summary
\end_layout
\end_inset
\end_layout
\begin_layout Standard
We have just seen the most basic ways of solving nonlinear unconstrained
optimization problems.
The online help system of Octave (try e.g.
\begin_inset Quotes eld
\end_inset
\family typewriter
help minimize
\family default
\begin_inset Quotes erd
\end_inset
) will yield information on other issues, such as
\emph on
passing extra arguments
\emph default
to the minimized function,
\emph on
controling the termination
\emph default
of the optimization process, choosing the algorithm etc.
\end_layout
\begin_layout LyX-Code
\end_layout
\end_body
\end_document
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