math::optimize - Optimisation routines

package require Tcl 8.2 package require math::optimize ?0.2? ::math::optimize::minimize begin end func maxerr ::math::optimize::maximize 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 constraints objective

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)

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.

Note: The linear programming algorithm is described but not yet operational.

This package defines the following public procedures:

::math::optimize::minimize 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::maximize 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 constraints objective

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") 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. objective - The M coefficients of the objective function

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]
    }

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 { { 1.0 1.0 1.0 } { 2.0 5.0 10.0 } } { 3.0 2.0 }]

Note, that a constraint like:

   x + y >= 1

   -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.

linear program, math, maximum, minimum, optimization

Copyright (c) 2004 Arjen Markus <arjenmarkus@users.sourceforge.net>
Copyright (c) 2004 Kevn B. Kenny <kennykb@users.sourceforge.net>