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#
# math/optimize.tcl --
#
# This file contains functions for optimization of a function
# or expression.
#
# Copyright (c) 2004, by Arjen Markus.
# Copyright (c) 2004, 2005 by Kevin B. Kenny. All rights reserved.
#
# See the file "license.terms" for information on usage and redistribution
# of this file, and for a DISCLAIMER OF ALL WARRANTIES.
#
# RCS: @(#) $Id: optimize.tcl,v 1.12 2011/01/18 07:49:53 arjenmarkus Exp $
#
#----------------------------------------------------------------------
package require Tcl 8.4
# math::optimize --
# Namespace for the commands
#
namespace eval ::math::optimize {
namespace export minimum maximum solveLinearProgram linearProgramMaximum
namespace export min_bound_1d min_unbound_1d
# Possible extension: minimumExpr, maximumExpr
}
# minimum --
# Minimize a given function over a given interval
#
# Arguments:
# begin Start of the interval
# end End of the interval
# func Name of the function to be minimized (takes one
# argument)
# maxerr Maximum relative error (defaults to 1.0e-4)
# Return value:
# Computed value for which the function is minimal
# Notes:
# The function needs not to be differentiable, but it is supposed
# to be continuous. There is no provision for sub-intervals where
# the function is constant (this might happen when the maximum
# error is very small, < 1.0e-15)
#
# Warning:
# This procedure is deprecated - use min_bound_1d instead
#
proc ::math::optimize::minimum { begin end func {maxerr 1.0e-4} } {
set nosteps [expr {3+int(-log($maxerr)/log(2.0))}]
set delta [expr {0.5*($end-$begin)*$maxerr}]
for { set step 0 } { $step < $nosteps } { incr step } {
set x1 [expr {($end+$begin)/2.0}]
set x2 [expr {$x1+$delta}]
set fx1 [uplevel 1 $func $x1]
set fx2 [uplevel 1 $func $x2]
if {$fx1 < $fx2} {
set end $x1
} else {
set begin $x1
}
}
return $x1
}
# maximum --
# Maximize a given function over a given interval
#
# Arguments:
# begin Start of the interval
# end End of the interval
# func Name of the function to be maximized (takes one
# argument)
# maxerr Maximum relative error (defaults to 1.0e-4)
# Return value:
# Computed value for which the function is maximal
# Notes:
# The function needs not to be differentiable, but it is supposed
# to be continuous. There is no provision for sub-intervals where
# the function is constant (this might happen when the maximum
# error is very small, < 1.0e-15)
#
# Warning:
# This procedure is deprecated - use max_bound_1d instead
#
proc ::math::optimize::maximum { begin end func {maxerr 1.0e-4} } {
set nosteps [expr {3+int(-log($maxerr)/log(2.0))}]
set delta [expr {0.5*($end-$begin)*$maxerr}]
for { set step 0 } { $step < $nosteps } { incr step } {
set x1 [expr {($end+$begin)/2.0}]
set x2 [expr {$x1+$delta}]
set fx1 [uplevel 1 $func $x1]
set fx2 [uplevel 1 $func $x2]
if {$fx1 > $fx2} {
set end $x1
} else {
set begin $x1
}
}
return $x1
}
#----------------------------------------------------------------------
#
# min_bound_1d --
#
# Find a local minimum of a function between two given
# abscissae. Derivative of f is not required.
#
# Usage:
# min_bound_1d f x1 x2 ?-option value?,,,
#
# Parameters:
# f - Function to minimize. Must be expressed as a Tcl
# command, to which will be appended the value at which
# to evaluate the function.
# x1 - Lower bound of the interval in which to search for a
# minimum
# x2 - Upper bound of the interval in which to search for a minimum
#
# Options:
# -relerror value
# Gives the tolerance desired for the returned
# abscissa. Default is 1.0e-7. Should never be less
# than the square root of the machine precision.
# -maxiter n
# Constrains minimize_bound_1d to evaluate the function
# no more than n times. Default is 100. If convergence
# is not achieved after the specified number of iterations,
# an error is thrown.
# -guess value
# Gives a point between x1 and x2 that is an initial guess
# for the minimum. f(guess) must be at most f(x1) or
# f(x2).
# -fguess value
# Gives the value of the ordinate at the value of '-guess'
# if known. Default is to evaluate the function
# -abserror value
# Gives the desired absolute error for the returned
# abscissa. Default is 1.0e-10.
# -trace boolean
# A true value causes a trace to the standard output
# of the function evaluations. Default is 0.
#
# Results:
# Returns a two-element list comprising the abscissa at which
# the function reaches a local minimum within the interval,
# and the value of the function at that point.
#
# Side effects:
# Whatever side effects arise from evaluating the given function.
#
#----------------------------------------------------------------------
proc ::math::optimize::min_bound_1d { f x1 x2 args } {
set f [lreplace $f 0 0 [uplevel 1 [list namespace which [lindex $f 0]]]]
set phim1 0.6180339887498949
set twomphi 0.3819660112501051
array set params {
-relerror 1.0e-7
-abserror 1.0e-10
-maxiter 100
-trace 0
-fguess {}
}
set params(-guess) [expr { $phim1 * $x1 + $twomphi * $x2 }]
if { ( [llength $args] % 2 ) != 0 } {
return -code error -errorcode [list min_bound_1d wrongNumArgs] \
"wrong \# args, should be\
\"[lreplace [info level 0] 1 end f x1 x2 ?-option value?...]\""
}
foreach { key value } $args {
if { ![info exists params($key)] } {
return -code error -errorcode [list min_bound_1d badoption $key] \
"unknown option \"$key\",\
should be -abserror,\
-fguess, -guess, -initial, -maxiter, -relerror,\
or -trace"
}
set params($key) $value
}
# a and b presumably bracket the minimum of the function. Make sure
# they're in ascending order.
if { $x1 < $x2 } {
set a $x1; set b $x2
} else {
set b $x1; set a $x2
}
set x $params(-guess); # Best abscissa found so far
set w $x; # Second best abscissa found so far
set v $x; # Most recent earlier value of w
set e 0.0; # Distance moved on the step before
# last.
# Evaluate the function at the initial guess
if { $params(-fguess) ne {} } {
set fx $params(-fguess)
} else {
set s $f; lappend s $x; set fx [eval $s]
if { $params(-trace) } {
puts stdout "f($x) = $fx (initialisation)"
}
}
set fw $fx
set fv $fx
for { set iter 0 } { $iter < $params(-maxiter) } { incr iter } {
# Find the midpoint of the current interval
set xm [expr { 0.5 * ( $a + $b ) }]
# Compute the current tolerance for x, and twice its value
set tol [expr { $params(-relerror) * abs($x) + $params(-abserror) }]
set tol2 [expr { $tol + $tol }]
if { abs( $x - $xm ) <= $tol2 - 0.5 * ($b - $a) } {
return [list $x $fx]
}
set golden 1
if { abs($e) > $tol } {
# Use parabolic interpolation to find a minimum determined
# by the evaluations at x, v, and w. The size of the step
# to take will be $p/$q.
set r [expr { ( $x - $w ) * ( $fx - $fv ) }]
set q [expr { ( $x - $v ) * ( $fx - $fw ) }]
set p [expr { ( $x - $v ) * $q - ( $x - $w ) * $r }]
set q [expr { 2. * ( $q - $r ) }]
if { $q > 0 } {
set p [expr { - $p }]
} else {
set q [expr { - $q }]
}
set olde $e
set e $d
# Test if parabolic interpolation results in less than half
# the movement of the step two steps ago.
if { abs($p) < abs( .5 * $q * $olde )
&& $p > $q * ( $a - $x )
&& $p < $q * ( $b - $x ) } {
set d [expr { $p / $q }]
set u [expr { $x + $d }]
if { ( $u - $a ) < $tol2 || ( $b - $u ) < $tol2 } {
if { $xm-$x < 0 } {
set d [expr { - $tol }]
} else {
set d $tol
}
}
set golden 0
}
}
# If parabolic interpolation didn't come up with an acceptable
# result, use Golden Section instead.
if { $golden } {
if { $x >= $xm } {
set e [expr { $a - $x }]
} else {
set e [expr { $b - $x }]
}
set d [expr { $twomphi * $e }]
}
# At this point, d is the size of the step to take. Make sure
# that it's at least $tol.
if { abs($d) >= $tol } {
set u [expr { $x + $d }]
} elseif { $d < 0 } {
set u [expr { $x - $tol }]
} else {
set u [expr { $x + $tol }]
}
# Evaluate the function
set s $f; lappend s $u; set fu [eval $s]
if { $params(-trace) } {
if { $golden } {
puts stdout "f($u)=$fu (golden section)"
} else {
puts stdout "f($u)=$fu (parabolic interpolation)"
}
}
if { $fu <= $fx } {
# We've the best abscissa so far.
if { $u >= $x } {
set a $x
} else {
set b $x
}
set v $w
set fv $fw
set w $x
set fw $fx
set x $u
set fx $fu
} else {
if { $u < $x } {
set a $u
} else {
set b $u
}
if { $fu <= $fw || $w == $x } {
# We've the second-best abscissa so far
set v $w
set fv $fw
set w $u
set fw $fu
} elseif { $fu <= $fv || $v == $x || $v == $w } {
# We've the third-best so far
set v $u
set fv $fu
}
}
}
return -code error -errorcode [list min_bound_1d noconverge $iter] \
"[lindex [info level 0] 0] failed to converge after $iter steps."
}
#----------------------------------------------------------------------
#
# brackmin --
#
# Find a place along the number line where a given function has
# a local minimum.
#
# Usage:
# brackmin f x1 x2 ?trace?
#
# Parameters:
# f - Function to minimize
# x1 - Abscissa thought to be near the minimum
# x2 - Additional abscissa thought to be near the minimum
# trace - Boolean variable that, if true,
# causes 'brackmin' to print a trace of its function
# evaluations to the standard output. Default is 0.
#
# Results:
# Returns a three element list {x1 y1 x2 y2 x3 y3} where
# y1=f(x1), y2=f(x2), y3=f(x3). x2 lies between x1 and x3, and
# y1>y2, y3>y2, proving that there is a local minimum somewhere
# in the interval (x1,x3).
#
# Side effects:
# Whatever effects the evaluation of f has.
#
#----------------------------------------------------------------------
proc ::math::optimize::brackmin { f x1 x2 {trace 0} } {
set f [lreplace $f 0 0 [uplevel 1 [list namespace which [lindex $f 0]]]]
set phi 1.6180339887498949
set epsilon 1.0e-20
set limit 50.
# Choose a and b so that f(a) < f(b)
set cmd $f; lappend cmd $x1; set fx1 [eval $cmd]
if { $trace } {
puts "f($x1) = $fx1 (initialisation)"
}
set cmd $f; lappend cmd $x2; set fx2 [eval $cmd]
if { $trace } {
puts "f($x2) = $fx2 (initialisation)"
}
if { $fx1 > $fx2 } {
set a $x1; set fa $fx1
set b $x2; set fb $fx2
} else {
set a $x2; set fa $fx2
set b $x1; set fb $fx1
}
# Choose a c in the downhill direction
set c [expr { $b + $phi * ($b - $a) }]
set cmd $f; lappend cmd $c; set fc [eval $cmd]
if { $trace } {
puts "f($c) = $fc (initial dilatation by phi)"
}
while { $fb >= $fc } {
# Try to do parabolic extrapolation to the minimum
set r [expr { ($b - $a) * ($fb - $fc) }]
set q [expr { ($b - $c) * ($fb - $fa) }]
if { abs( $q - $r ) > $epsilon } {
set denom [expr { $q - $r }]
} elseif { $q > $r } {
set denom $epsilon
} else {
set denom -$epsilon
}
set u [expr { $b - ( (($b - $c) * $q - ($b - $a) * $r)
/ (2. * $denom) ) }]
set ulimit [expr { $b + $limit * ( $c - $b ) }]
# Test the extrapolated abscissa
if { ($b - $u) * ($u - $c) > 0 } {
# u lies between b and c. Try to interpolate
set cmd $f; lappend cmd $u; set fu [eval $cmd]
if { $trace } {
puts "f($u) = $fu (parabolic interpolation)"
}
if { $fu < $fc } {
# fb > fu and fc > fu, so there is a minimum between b and c
# with u as a starting guess.
return [list $b $fb $u $fu $c $fc]
}
if { $fu > $fb } {
# fb < fu, fb < fa, and u cannot lie between a and b
# (because it lies between a and c). There is a minimum
# somewhere between a and u, with b a starting guess.
return [list $a $fa $b $fb $u $fu]
}
# Parabolic interpolation was useless. Expand the
# distance by a factor of phi and try again.
set u [expr { $c + $phi * ($c - $b) }]
set cmd $f; lappend cmd $u; set fu [eval $cmd]
if { $trace } {
puts "f($u) = $fu (parabolic interpolation failed)"
}
} elseif { ( $c - $u ) * ( $u - $ulimit ) > 0 } {
# u lies between $c and $ulimit.
set cmd $f; lappend cmd $u; set fu [eval $cmd]
if { $trace } {
puts "f($u) = $fu (parabolic extrapolation)"
}
if { $fu > $fc } {
# minimum lies between b and u, with c an initial guess.
return [list $b $fb $c $fc $u $fu]
}
# function is still decreasing fa > fb > fc > fu. Take
# another factor-of-phi step.
set b $c; set fb $fc
set c $u; set fc $fu
set u [expr { $c + $phi * ( $c - $b ) }]
set cmd $f; lappend cmd $u; set fu [eval $cmd]
if { $trace } {
puts "f($u) = $fu (parabolic extrapolation ok)"
}
} elseif { ($u - $ulimit) * ( $ulimit - $c ) >= 0 } {
# u went past ulimit. Pull in to ulimit and evaluate there.
set u $ulimit
set cmd $f; lappend cmd $u; set fu [eval $cmd]
if { $trace } {
puts "f($u) = $fu (limited step)"
}
} else {
# parabolic extrapolation gave a useless value.
set u [expr { $c + $phi * ( $c - $b ) }]
set cmd $f; lappend cmd $u; set fu [eval $cmd]
if { $trace } {
puts "f($u) = $fu (parabolic extrapolation failed)"
}
}
set a $b; set fa $fb
set b $c; set fb $fc
set c $u; set fc $fu
}
return [list $a $fa $b $fb $c $fc]
}
#----------------------------------------------------------------------
#
# min_unbound_1d --
#
# Minimize a function of one variable, unconstrained, derivatives
# not required.
#
# Usage:
# min_bound_1d f x1 x2 ?-option value?,,,
#
# Parameters:
# f - Function to minimize. Must be expressed as a Tcl
# command, to which will be appended the value at which
# to evaluate the function.
# x1 - Initial guess at the minimum
# x2 - Second initial guess at the minimum, used to set the
# initial length scale for the search.
#
# Options:
# -relerror value
# Gives the tolerance desired for the returned
# abscissa. Default is 1.0e-7. Should never be less
# than the square root of the machine precision.
# -maxiter n
# Constrains min_bound_1d to evaluate the function
# no more than n times. Default is 100. If convergence
# is not achieved after the specified number of iterations,
# an error is thrown.
# -abserror value
# Gives the desired absolute error for the returned
# abscissa. Default is 1.0e-10.
# -trace boolean
# A true value causes a trace to the standard output
# of the function evaluations. Default is 0.
#
#----------------------------------------------------------------------
proc ::math::optimize::min_unbound_1d { f x1 x2 args } {
set f [lreplace $f 0 0 [uplevel 1 [list namespace which [lindex $f 0]]]]
array set params {
-relerror 1.0e-7
-abserror 1.0e-10
-maxiter 100
-trace 0
}
if { ( [llength $args] % 2 ) != 0 } {
return -code error -errorcode [list min_unbound_1d wrongNumArgs] \
"wrong \# args, should be\
\"[lreplace [info level 0] 1 end \
f x1 x2 ?-option value?...]\""
}
foreach { key value } $args {
if { ![info exists params($key)] } {
return -code error -errorcode [list min_unbound_1d badoption $key] \
"unknown option \"$key\",\
should be -trace"
}
set params($key) $value
}
foreach { a fa b fb c fc } [brackmin $f $x1 $x2 $params(-trace)] {
break
}
return [eval [linsert [array get params] 0 \
min_bound_1d $f $a $c -guess $b -fguess $fb]]
}
#----------------------------------------------------------------------
#
# nelderMead --
#
# Attempt to minimize/maximize a function using the downhill
# simplex method of Nelder and Mead.
#
# Usage:
# nelderMead f x ?-keyword value?
#
# Parameters:
# f - The function to minimize. The function must be an incomplete
# Tcl command, to which will be appended N parameters.
# x - The starting guess for the minimum; a vector of N parameters
# to be passed to the function f.
#
# Options:
# -scale xscale
# Initial guess as to the problem scale. If '-scale' is
# supplied, then the parameters will be varied by the
# specified amounts. The '-scale' parameter must of the
# same dimension as the 'x' vector, and all elements must
# be nonzero. Default is 0.0001 times the 'x' vector,
# or 0.0001 for zero elements in the 'x' vector.
#
# -ftol epsilon
# Requested tolerance in the function value; nelderMead
# returns if N+1 consecutive iterates all differ by less
# than the -ftol value. Default is 1.0e-7
#
# -maxiter N
# Maximum number of iterations to attempt. Default is
# 500.
#
# -trace flag
# If '-trace 1' is supplied, nelderMead writes a record
# of function evaluations to the standard output as it
# goes. Default is 0.
#
#----------------------------------------------------------------------
proc ::math::optimize::nelderMead { f startx args } {
array set params {
-ftol 1.e-7
-maxiter 500
-scale {}
-trace 0
}
# Check arguments
if { ( [llength $args] % 2 ) != 0 } {
return -code error -errorcode [list nelderMead wrongNumArgs] \
"wrong \# args, should be\
\"[lreplace [info level 0] 1 end \
f x1 x2 ?-option value?...]\""
}
foreach { key value } $args {
if { ![info exists params($key)] } {
return -code error -errorcode [list nelderMead badoption $key] \
"unknown option \"$key\",\
should be -ftol, -maxiter, -scale or -trace"
}
set params($key) $value
}
# Construct the initial simplex
set vertices [list $startx]
if { [llength $params(-scale)] == 0 } {
set i 0
foreach x0 $startx {
if { $x0 == 0 } {
set x1 0.0001
} else {
set x1 [expr {1.0001 * $x0}]
}
lappend vertices [lreplace $startx $i $i $x1]
incr i
}
} elseif { [llength $params(-scale)] != [llength $startx] } {
return -code error -errorcode [list nelderMead badOption -scale] \
"-scale vector must be of same size as starting x vector"
} else {
set i 0
foreach x0 $startx s $params(-scale) {
lappend vertices [lreplace $startx $i $i [expr { $x0 + $s }]]
incr i
}
}
# Evaluate at the initial points
set n [llength $startx]
foreach x $vertices {
set cmd $f
foreach xx $x {
lappend cmd $xx
}
set y [uplevel 1 $cmd]
if {$params(-trace)} {
puts "nelderMead: evaluating initial point: x=[list $x] y=$y"
}
lappend yvec $y
}
# Loop adjusting the simplex in the 'vertices' array.
set nIter 0
while { 1 } {
# Find the highest, next highest, and lowest value in y,
# and save the indices.
set iBot 0
set yBot [lindex $yvec 0]
set iTop -1
set yTop [lindex $yvec 0]
set iNext -1
set i 0
foreach y $yvec {
if { $y <= $yBot } {
set yBot $y
set iBot $i
}
if { $iTop < 0 || $y >= $yTop } {
set iNext $iTop
set yNext $yTop
set iTop $i
set yTop $y
} elseif { $iNext < 0 || $y >= $yNext } {
set iNext $i
set yNext $y
}
incr i
}
# Return if the relative error is within an acceptable range
set rerror [expr { 2. * abs( $yTop - $yBot )
/ ( abs( $yTop ) + abs( $yBot ) + $params(-ftol) ) }]
if { $rerror < $params(-ftol) } {
set status ok
break
}
# Count iterations
if { [incr nIter] > $params(-maxiter) } {
set status too-many-iterations
break
}
incr nIter
# Find the centroid of the face opposite the vertex that
# maximizes the function value.
set centroid {}
for { set i 0 } { $i < $n } { incr i } {
lappend centroid 0.0
}
set i 0
foreach v $vertices {
if { $i != $iTop } {
set newCentroid {}
foreach x0 $centroid x1 $v {
lappend newCentroid [expr { $x0 + $x1 }]
}
set centroid $newCentroid
}
incr i
}
set newCentroid {}
foreach x $centroid {
lappend newCentroid [expr { $x / $n }]
}
set centroid $newCentroid
# The first trial point is a reflection of the high point
# around the centroid
set trial {}
foreach x0 [lindex $vertices $iTop] x1 $centroid {
lappend trial [expr {$x1 + ($x1 - $x0)}]
}
set cmd $f
foreach xx $trial {
lappend cmd $xx
}
set yTrial [uplevel 1 $cmd]
if { $params(-trace) } {
puts "nelderMead: trying reflection: x=[list $trial] y=$yTrial"
}
# If that reflection yields a new minimum, replace the high point,
# and additionally try dilating in the same direction.
if { $yTrial < $yBot } {
set trial2 {}
foreach x0 $centroid x1 $trial {
lappend trial2 [expr { $x1 + ($x1 - $x0) }]
}
set cmd $f
foreach xx $trial2 {
lappend cmd $xx
}
set yTrial2 [uplevel 1 $cmd]
if { $params(-trace) } {
puts "nelderMead: trying dilated reflection:\
x=[list $trial2] y=$y"
}
if { $yTrial2 < $yBot } {
# Additional dilation yields a new minimum
lset vertices $iTop $trial2
lset yvec $iTop $yTrial2
} else {
# Additional dilation failed, but we can still use
# the first trial point.
lset vertices $iTop $trial
lset yvec $iTop $yTrial
}
} elseif { $yTrial < $yNext } {
# The reflected point isn't a new minimum, but it's
# better than the second-highest. Replace the old high
# point and try again.
lset vertices $iTop $trial
lset yvec $iTop $yTrial
} else {
# The reflected point is worse than the second-highest point.
# If it's better than the highest, keep it... but in any case,
# we want to try contracting the simplex, because a further
# reflection will simply bring us back to the starting point.
if { $yTrial < $yTop } {
lset vertices $iTop $trial
lset yvec $iTop $yTrial
set yTop $yTrial
}
set trial {}
foreach x0 [lindex $vertices $iTop] x1 $centroid {
lappend trial [expr { ( $x0 + $x1 ) / 2. }]
}
set cmd $f
foreach xx $trial {
lappend cmd $xx
}
set yTrial [uplevel 1 $cmd]
if { $params(-trace) } {
puts "nelderMead: contracting from high point:\
x=[list $trial] y=$y"
}
if { $yTrial < $yTop } {
# Contraction gave an improvement, so continue with
# the smaller simplex
lset vertices $iTop $trial
lset yvec $iTop $yTrial
} else {
# Contraction gave no improvement either; we seem to
# be in a valley of peculiar topology. Contract the
# simplex about the low point and try again.
set newVertices {}
set newYvec {}
set i 0
foreach v $vertices y $yvec {
if { $i == $iBot } {
lappend newVertices $v
lappend newYvec $y
} else {
set newv {}
foreach x0 $v x1 [lindex $vertices $iBot] {
lappend newv [expr { ($x0 + $x1) / 2. }]
}
lappend newVertices $newv
set cmd $f
foreach xx $newv {
lappend cmd $xx
}
lappend newYvec [uplevel 1 $cmd]
if { $params(-trace) } {
puts "nelderMead: contracting about low point:\
x=[list $newv] y=$y"
}
}
incr i
}
set vertices $newVertices
set yvec $newYvec
}
}
}
return [list y $yBot x [lindex $vertices $iBot] vertices $vertices yvec $yvec nIter $nIter status $status]
}
# solveLinearProgram
# Solve a linear program in standard form
#
# Arguments:
# objective Vector defining the objective function
# constraints Matrix of constraints (as a list of lists)
#
# Return value:
# Computed values for the coordinates or "unbounded" or "infeasible"
#
proc ::math::optimize::solveLinearProgram { objective constraints } {
#
# Check the arguments first and then put them in a more convenient
# form
#
foreach {nconst nvars matrix} \
[SimplexPrepareMatrix $objective $constraints] {break}
set solution [SimplexSolve $nconst nvars $matrix]
if { [llength $solution] > 1 } {
return [lrange $solution 0 [expr {$nvars-1}]]
} else {
return $solution
}
}
# linearProgramMaximum --
# Compute the value attained at the optimum
#
# Arguments:
# objective The coefficients of the objective function
# result The coordinate values as obtained by solving the program
#
# Return value:
# Value at the maximum point
#
proc ::math::optimize::linearProgramMaximum {objective result} {
set value 0.0
foreach coeff $objective coord $result {
set value [expr {$value+$coeff*$coord}]
}
return $value
}
# SimplexPrintMatrix
# Debugging routine: print the matrix in easy to read form
#
# Arguments:
# matrix Matrix to be printed
#
# Return value:
# None
#
# Note:
# The tableau should be transposed ...
#
proc ::math::optimize::SimplexPrintMatrix {matrix} {
puts "\nBasis:\t[join [lindex $matrix 0] \t]"
foreach col [lrange $matrix 1 end] {
puts " \t[join $col \t]"
}
}
# SimplexPrepareMatrix
# Prepare the standard tableau from all program data
#
# Arguments:
# objective Vector defining the objective function
# constraints Matrix of constraints (as a list of lists)
#
# Return value:
# List of values as a standard tableau and two values
# for the sizes
#
proc ::math::optimize::SimplexPrepareMatrix {objective constraints} {
#
# Check the arguments first
#
set nconst [llength $constraints]
set ncols {}
foreach row $constraints {
if { $ncols == {} } {
set ncols [llength $row]
} else {
if { $ncols != [llength $row] } {
return -code error -errorcode ARGS "Incorrectly formed constraints matrix"
}
}
}
set nvars [expr {$ncols-1}]
if { [llength $objective] != $nvars } {
return -code error -errorcode ARGS "Incorrect length for objective vector"
}
#
# Set up the tableau:
# Easiest manipulations if we store the columns first
# So:
# - First column is the list of variable indices in the basis
# - Second column is the list of maximum values
# - "nvars" columns that follow: the coefficients for the actual
# variables
# - last "nconst" columns: the slack variables
#
set matrix [list]
set lastrow [concat $objective [list 0.0]]
set newcol [list]
for {set idx 0} {$idx < $nconst} {incr idx} {
lappend newcol [expr {$nvars+$idx}]
}
lappend newcol "?"
lappend matrix $newcol
set zvector [list]
foreach row $constraints {
lappend zvector [lindex $row end]
}
lappend zvector 0.0
lappend matrix $zvector
for {set idx 0} {$idx < $nvars} {incr idx} {
set newcol [list]
foreach row $constraints {
lappend newcol [expr {double([lindex $row $idx])}]
}
lappend newcol [expr {-double([lindex $lastrow $idx])}]
lappend matrix $newcol
}
#
# Add the columns for the slack variables
#
set zeros {}
for {set idx 0} {$idx <= $nconst} {incr idx} {
lappend zeros 0.0
}
for {set idx 0} {$idx < $nconst} {incr idx} {
lappend matrix [lreplace $zeros $idx $idx 1.0]
}
return [list $nconst $nvars $matrix]
}
# SimplexSolve --
# Solve the given linear program using the simplex method
#
# Arguments:
# nconst Number of constraints
# nvars Number of actual variables
# tableau Standard tableau (as a list of columns)
#
# Return value:
# List of values for the actual variables
#
proc ::math::optimize::SimplexSolve {nconst nvars tableau} {
set end 0
while { !$end } {
#
# Find the new variable to put in the basis
#
set nextcol [SimplexFindNextColumn $tableau]
if { $nextcol == -1 } {
set end 1
continue
}
#
# Now determine which one should leave
# TODO: is a lack of a proper row indeed an
# indication of the infeasibility?
#
set nextrow [SimplexFindNextRow $tableau $nextcol]
if { $nextrow == -1 } {
return "unbounded"
}
#
# Make the vector for sweeping through the tableau
#
set vector [SimplexMakeVector $tableau $nextcol $nextrow]
#
# Sweep through the tableau
#
set tableau [SimplexNewTableau $tableau $nextcol $nextrow $vector]
}
#
# Now we can return the result
#
SimplexResult $tableau
}
# SimplexResult --
# Reconstruct the result vector
#
# Arguments:
# tableau Standard tableau (as a list of columns)
#
# Return value:
# Vector of values representing the maximum point
#
proc ::math::optimize::SimplexResult {tableau} {
set result {}
set firstcol [lindex $tableau 0]
set secondcol [lindex $tableau 1]
set result {}
set nvars [expr {[llength $tableau]-2}]
for {set i 0} {$i < $nvars } { incr i } {
lappend result 0.0
}
set idx 0
foreach col [lrange $firstcol 0 end-1] {
set value [lindex $secondcol $idx]
if { $value >= 0.0 } {
set result [lreplace $result $col $col [lindex $secondcol $idx]]
incr idx
} else {
# If a negative component, then the problem was not feasible
return "infeasible"
}
}
return $result
}
# SimplexFindNextColumn --
# Find the next column - the one with the largest negative
# coefficient
#
# Arguments:
# tableau Standard tableau (as a list of columns)
#
# Return value:
# Index of the column
#
proc ::math::optimize::SimplexFindNextColumn {tableau} {
set idx 0
set minidx -1
set mincoeff 0.0
foreach col [lrange $tableau 2 end] {
set coeff [lindex $col end]
if { $coeff < 0.0 } {
if { $coeff < $mincoeff } {
set minidx $idx
set mincoeff $coeff
}
}
incr idx
}
return $minidx
}
# SimplexFindNextRow --
# Find the next row - the one with the largest negative
# coefficient
#
# Arguments:
# tableau Standard tableau (as a list of columns)
# nextcol Index of the variable that will replace this one
#
# Return value:
# Index of the row
#
proc ::math::optimize::SimplexFindNextRow {tableau nextcol} {
set idx 0
set minidx -1
set mincoeff {}
set bvalues [lrange [lindex $tableau 1] 0 end-1]
set yvalues [lrange [lindex $tableau [expr {2+$nextcol}]] 0 end-1]
foreach rowcoeff $bvalues divcoeff $yvalues {
if { $divcoeff > 0.0 } {
set coeff [expr {$rowcoeff/$divcoeff}]
if { $mincoeff == {} || $coeff < $mincoeff } {
set minidx $idx
set mincoeff $coeff
}
}
incr idx
}
return $minidx
}
# SimplexMakeVector --
# Make the "sweep" vector
#
# Arguments:
# tableau Standard tableau (as a list of columns)
# nextcol Index of the variable that will replace this one
# nextrow Index of the variable in the base that will be replaced
#
# Return value:
# Vector to be used to update the coefficients of the tableau
#
proc ::math::optimize::SimplexMakeVector {tableau nextcol nextrow} {
set idx 0
set vector {}
set column [lindex $tableau [expr {2+$nextcol}]]
set divcoeff [lindex $column $nextrow]
foreach colcoeff $column {
if { $idx != $nextrow } {
set coeff [expr {-$colcoeff/$divcoeff}]
} else {
set coeff [expr {1.0/$divcoeff-1.0}]
}
lappend vector $coeff
incr idx
}
return $vector
}
# SimplexNewTableau --
# Sweep through the tableau and create the new one
#
# Arguments:
# tableau Standard tableau (as a list of columns)
# nextcol Index of the variable that will replace this one
# nextrow Index of the variable in the base that will be replaced
# vector Vector to sweep with
#
# Return value:
# New tableau
#
proc ::math::optimize::SimplexNewTableau {tableau nextcol nextrow vector} {
#
# The first column: replace the nextrow-th element
# The second column: replace the value at the nextrow-th element
# For all the others: the same receipe
#
set firstcol [lreplace [lindex $tableau 0] $nextrow $nextrow $nextcol]
set newtableau [list $firstcol]
#
# The rest of the matrix
#
foreach column [lrange $tableau 1 end] {
set yval [lindex $column $nextrow]
set newcol {}
foreach c $column vcoeff $vector {
set newval [expr {$c+$yval*$vcoeff}]
lappend newcol $newval
}
lappend newtableau $newcol
}
return $newtableau
}
# Now we can announce our presence
package provide math::optimize 1.0.1
if { ![info exists ::argv0] || [string compare $::argv0 [info script]] } {
return
}
namespace import math::optimize::min_bound_1d
namespace import math::optimize::maximum
namespace import math::optimize::nelderMead
proc f {x y} {
set xx [expr { $x - 3.1415926535897932 / 2. }]
set v1 [expr { 0.3 * exp( -$xx*$xx / 2. ) }]
set d [expr { 10. * $y - sin(9. * $x) }]
set v2 [expr { exp(-10.*$d*$d)}]
set rv [expr { -$v1 - $v2 }]
return $rv
}
proc g {a b} {
set x1 [expr {0.1 - $a + $b}]
set x2 [expr {$a + $b - 1.}]
set x3 [expr {3.-8.*$a+8.*$a*$a-8.*$b+8.*$b*$b}]
set x4 [expr {$a/10. + $b/10. + $x1*$x1/3. + $x2*$x2 - $x2 * exp(1-$x3*$x3)}]
return $x4
}
set prec $::tcl_precision
if {![package vsatisfies [package provide Tcl] 8.5]} {
set ::tcl_precision 17
} else {
set ::tcl_precision 0
}
puts "f"
puts [math::optimize::nelderMead f {1. 0.} -scale {0.1 0.01} -trace 1]
puts "g"
puts [math::optimize::nelderMead g {0. 0.} -scale {1. 1.} -trace 1]
set ::tcl_precision $prec
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