math::optimize(n)              Tcl Math Library              math::optimize(n)



______________________________________________________________________________

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:

       o      Minimize or maximize a function over a given interval

       o      Solve a linear program (maximize a linear function subject to
              linear constraints)

       o      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 SF Trackers
       [http://sourceforge.net/tracker/?group_id=12883].  Please also report
       any ideas for enhancements you may have for either package and/or
       documentation.

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>




math                                  1.0                    math::optimize(n)
