Provided by: tcllib_1.21+dfsg-1_all 
NAME
math::optimize - Optimisation routines
SYNOPSIS
package require Tcl 8.4
package require math::optimize ?1.0?
::math::optimize::minimum begin end func maxerr
::math::optimize::maximum begin end func maxerr
::math::optimize::min_bound_1d func begin end ?-relerror reltol? ?-abserror abstol? ?-maxiter maxiter?
?-trace traceflag?
::math::optimize::min_unbound_1d func begin end ?-relerror reltol? ?-abserror abstol? ?-maxiter maxiter?
?-trace traceflag?
::math::optimize::solveLinearProgram objective constraints
::math::optimize::linearProgramMaximum objective result
::math::optimize::nelderMead objective xVector ?-scale xScaleVector? ?-ftol epsilon? ?-maxiter count?
??-trace? flag?
________________________________________________________________________________________________________________
DESCRIPTION
This package implements several optimisation algorithms:
• Minimize or maximize a function over a given interval
• Solve a linear program (maximize a linear function subject to linear constraints)
• Minimize a function of several variables given an initial guess for the location of the minimum.
The package is fully implemented in Tcl. No particular attention has been paid to the accuracy of the
calculations. Instead, the algorithms have been used in a straightforward manner.
This document describes the procedures and explains their usage.
PROCEDURES
This package defines the following public procedures:
::math::optimize::minimum begin end func maxerr
Minimize the given (continuous) function by examining the values in the given interval. The
procedure determines the values at both ends and in the centre of the interval and then constructs
a new interval of 1/2 length that includes the minimum. No guarantee is made that the global
minimum is found.
The procedure returns the "x" value for which the function is minimal.
This procedure has been deprecated - use min_bound_1d instead
begin - Start of the interval
end - End of the interval
func - Name of the function to be minimized (a procedure taking one argument).
maxerr - Maximum relative error (defaults to 1.0e-4)
::math::optimize::maximum begin end func maxerr
Maximize the given (continuous) function by examining the values in the given interval. The
procedure determines the values at both ends and in the centre of the interval and then constructs
a new interval of 1/2 length that includes the maximum. No guarantee is made that the global
maximum is found.
The procedure returns the "x" value for which the function is maximal.
This procedure has been deprecated - use max_bound_1d instead
begin - Start of the interval
end - End of the interval
func - Name of the function to be maximized (a procedure taking one argument).
maxerr - Maximum relative error (defaults to 1.0e-4)
::math::optimize::min_bound_1d func begin end ?-relerror reltol? ?-abserror abstol? ?-maxiter maxiter?
?-trace traceflag?
Miminizes a function of one variable in the given interval. The procedure uses Brent's method of
parabolic interpolation, protected by golden-section subdivisions if the interpolation is not
converging. No guarantee is made that a global minimum is found. The function to evaluate, func,
must be a single Tcl command; it will be evaluated with an abscissa appended as the last argument.
x1 and x2 are the two bounds of the interval in which the minimum is to be found. They need not
be in increasing order.
reltol, if specified, is the desired upper bound on the relative error of the result; default is
1.0e-7. The given value should never be smaller than the square root of the machine's floating
point precision, or else convergence is not guaranteed. abstol, if specified, is the desired
upper bound on the absolute error of the result; default is 1.0e-10. Caution must be used with
small values of abstol to avoid overflow/underflow conditions; if the minimum is expected to lie
about a small but non-zero abscissa, you consider either shifting the function or changing its
length scale.
maxiter may be used to constrain the number of function evaluations to be performed; default is
100. If the command evaluates the function more than maxiter times, it returns an error to the
caller.
traceFlag is a Boolean value. If true, it causes the command to print a message on the standard
output giving the abscissa and ordinate at each function evaluation, together with an indication
of what type of interpolation was chosen. Default is 0 (no trace).
::math::optimize::min_unbound_1d func begin end ?-relerror reltol? ?-abserror abstol? ?-maxiter maxiter?
?-trace traceflag?
Miminizes a function of one variable over the entire real number line. The procedure uses
parabolic extrapolation combined with golden-section dilatation to search for a region where a
minimum exists, followed by Brent's method of parabolic interpolation, protected by golden-section
subdivisions if the interpolation is not converging. No guarantee is made that a global minimum
is found. The function to evaluate, func, must be a single Tcl command; it will be evaluated with
an abscissa appended as the last argument.
x1 and x2 are two initial guesses at where the minimum may lie. x1 is the starting point for the
minimization, and the difference between x2 and x1 is used as a hint at the characteristic length
scale of the problem.
reltol, if specified, is the desired upper bound on the relative error of the result; default is
1.0e-7. The given value should never be smaller than the square root of the machine's floating
point precision, or else convergence is not guaranteed. abstol, if specified, is the desired
upper bound on the absolute error of the result; default is 1.0e-10. Caution must be used with
small values of abstol to avoid overflow/underflow conditions; if the minimum is expected to lie
about a small but non-zero abscissa, you consider either shifting the function or changing its
length scale.
maxiter may be used to constrain the number of function evaluations to be performed; default is
100. If the command evaluates the function more than maxiter times, it returns an error to the
caller.
traceFlag is a Boolean value. If true, it causes the command to print a message on the standard
output giving the abscissa and ordinate at each function evaluation, together with an indication
of what type of interpolation was chosen. Default is 0 (no trace).
::math::optimize::solveLinearProgram objective constraints
Solve a linear program in standard form using a straightforward implementation of the Simplex
algorithm. (In the explanation below: The linear program has N constraints and M variables).
The procedure returns a list of M values, the values for which the objective function is maximal
or a single keyword if the linear program is not feasible or unbounded (either "unfeasible" or
"unbounded")
objective - The M coefficients of the objective function
constraints - Matrix of coefficients plus maximum values that implement the linear constraints. It
is expected to be a list of N lists of M+1 numbers each, M coefficients and the maximum value.
::math::optimize::linearProgramMaximum objective result
Convenience function to return the maximum for the solution found by the solveLinearProgram
procedure.
objective - The M coefficients of the objective function
result - The result as returned by solveLinearProgram
::math::optimize::nelderMead objective xVector ?-scale xScaleVector? ?-ftol epsilon? ?-maxiter count?
??-trace? flag?
Minimizes, in unconstrained fashion, a function of several variable over all of space. The
function to evaluate, objective, must be a single Tcl command. To it will be appended as many
elements as appear in the initial guess at the location of the minimum, passed in as a Tcl list,
xVector.
xScaleVector is an initial guess at the problem scale; the first function evaluations will be made
by varying the co-ordinates in xVector by the amounts in xScaleVector. If xScaleVector is not
supplied, the co-ordinates will be varied by a factor of 1.0001 (if the co-ordinate is non-zero)
or by a constant 0.0001 (if the co-ordinate is zero).
epsilon is the desired relative error in the value of the function evaluated at the minimum. The
default is 1.0e-7, which usually gives three significant digits of accuracy in the values of the
x's.
pp count is a limit on the number of trips through the main loop of the optimizer. The number of
function evaluations may be several times this number. If the optimizer fails to find a minimum
to within ftol in maxiter iterations, it returns its current best guess and an error status.
Default is to allow 500 iterations.
flag is a flag that, if true, causes a line to be written to the standard output for each
evaluation of the objective function, giving the arguments presented to the function and the value
returned. Default is false.
The nelderMead procedure returns a list of alternating keywords and values suitable for use with
array set. The meaning of the keywords is:
x is the approximate location of the minimum.
y is the value of the function at x.
yvec is a vector of the best N+1 function values achieved, where N is the dimension of x
vertices is a list of vectors giving the function arguments corresponding to the values in yvec.
nIter is the number of iterations required to achieve convergence or fail.
status is 'ok' if the operation succeeded, or 'too-many-iterations' if the maximum iteration count
was exceeded.
nelderMead minimizes the given function using the downhill simplex method of Nelder and Mead.
This method is quite slow - much faster methods for minimization are known - but has the advantage
of being extremely robust in the face of problems where the minimum lies in a valley of complex
topology.
nelderMead can occasionally find itself "stuck" at a point where it can make no further progress;
it is recommended that the caller run it at least a second time, passing as the initial guess the
result found by the previous call. The second run is usually very fast.
nelderMead can be used in some cases for constrained optimization. To do this, add a large value
to the objective function if the parameters are outside the feasible region. To work effectively
in this mode, nelderMead requires that the initial guess be feasible and usually requires that the
feasible region be convex.
NOTES
Several of the above procedures take the names of procedures as arguments. To avoid problems with the
visibility of these procedures, the fully-qualified name of these procedures is determined inside the
optimize routines. For the user this has only one consequence: the named procedure must be visible in the
calling procedure. For instance:
namespace eval ::mySpace {
namespace export calcfunc
proc calcfunc { x } { return $x }
}
#
# Use a fully-qualified name
#
namespace eval ::myCalc {
puts [min_bound_1d ::myCalc::calcfunc $begin $end]
}
#
# Import the name
#
namespace eval ::myCalc {
namespace import ::mySpace::calcfunc
puts [min_bound_1d calcfunc $begin $end]
}
The simple procedures minimum and maximum have been deprecated: the alternatives are much more flexible,
robust and require less function evaluations.
EXAMPLES
Let us take a few simple examples:
Determine the maximum of f(x) = x^3 exp(-3x), on the interval (0,10):
proc efunc { x } { expr {$x*$x*$x * exp(-3.0*$x)} }
puts "Maximum at: [::math::optimize::max_bound_1d efunc 0.0 10.0]"
The maximum allowed error determines the number of steps taken (with each step in the iteration the
interval is reduced with a factor 1/2). Hence, a maximum error of 0.0001 is achieved in approximately 14
steps.
An example of a linear program is:
Optimise the expression 3x+2y, where:
x >= 0 and y >= 0 (implicit constraints, part of the
definition of linear programs)
x + y <= 1 (constraints specific to the problem)
2x + 5y <= 10
This problem can be solved as follows:
set solution [::math::optimize::solveLinearProgram { 3.0 2.0 } { { 1.0 1.0 1.0 }
{ 2.0 5.0 10.0 } } ]
Note, that a constraint like:
x + y >= 1
can be turned into standard form using:
-x -y <= -1
The theory of linear programming is the subject of many a text book and the Simplex algorithm that is
implemented here is the best-known method to solve this type of problems, but it is not the only one.
BUGS, IDEAS, FEEDBACK
This document, and the package it describes, will undoubtedly contain bugs and other problems. Please
report such in the category math :: optimize of the Tcllib Trackers
[http://core.tcl.tk/tcllib/reportlist]. Please also report any ideas for enhancements you may have for
either package and/or documentation.
When proposing code changes, please provide unified diffs, i.e the output of diff -u.
Note further that attachments are strongly preferred over inlined patches. Attachments can be made by
going to the Edit form of the ticket immediately after its creation, and then using the left-most button
in the secondary navigation bar.
KEYWORDS
linear program, math, maximum, minimum, optimization
CATEGORY
Mathematics
COPYRIGHT
Copyright (c) 2004 Arjen Markus <arjenmarkus@users.sourceforge.net> Copyright (c) 2004,2005 Kevn B. Kenny <kennykb@users.sourceforge.net>