math::optimize - Optimisation routines
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.
This package defines the following public procedures:
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)
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)
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).
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).
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.
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
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.
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 $end] }
The simple procedures minimum and maximum have been deprecated: the alternatives are much more flexible, robust and require less function evaluations.
Let us take a few simple examples:
Determine the maximum of f(x) = x^3 exp(-3x), on the interval (0,10):
proc efunc { x } { expr {$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.
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 SF Trackers. Please also report any ideas for enhancements you may have for either package and/or documentation.
Mathematics
Copyright © 2004 Arjen Markus
<arjenmarkus@users.sourceforge.net>
Copyright © 2004,2005 Kevn B. Kenny
<kennykb@users.sourceforge.net>