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# probopt_diffev.tcl --
# Implementation of the differential probopt algorithm
# for optimising functions
#
# Note:
# The algorithm does not confine the points to the given
# hyper block - it is merely used to initialise it.
#
namespace eval ::math::probopt {}
# diffev --
# Optimise a function using the differential probopt algorithm
#
# Arguments:
# func Function for which the global minimum is to be found
# bounds Boundaries for all independent variables of the function,
# as a list of pairs of minimum and maximum
# args Set of options - key-value pairs
#
# Result:
# Estimate of the global minimum as found via the procedure
#
proc ::math::probopt::diffev {func bounds args} {
#
# Set the default options
#
set dims [llength $bounds]
set options [dict create -number 0 -factor 0.6 -lambda 0.0 -crossover 0.5 \
-iterations 100 -maxevaluations 1.0e9 -abstolerance 0.0 -reltolerance 0.001]
#
# Handle the options
#
foreach {key value} $args {
if { [dict exists $options $key] } {
dict set options $key $value
} else {
return -code error "Unknown option: $key"
}
}
dict with options {}
if { ${-number} == 0 } {
set -number [expr {4 * $dims}]
dict set options -number ${-number}
}
#
# Set up the initial collections of points
#
set evals 0
set points {}
for {set i 0} {$i < ${-number}} {incr i} {
set coords [GeneratePoint $bounds]
lappend points [list $coords [$func $coords]]
incr evals
}
#puts [join $points \n]
#
# Iteration over the generations:
# - For each point, construct a new estimate and check if it is better
# - If it is, replace the original point by the new point
#
set oldIndex [IndexBestPoint $points]
set oldValue [lindex $points $oldIndex 1]
set bestPerGeneration {}
for {set generation 0} {$generation < ${-iterations}} {incr generation} {
#puts "$generation"
set newPoints {}
set renewed 0 ;# Keep track of the replacement of points to avoid
# a premature ending
for {set i 0} {$i < ${-number}} {incr i} {
set point [lindex $points $i]
set newCoords [ConstructNewCoords $points ${-factor} ${-lambda} ${-crossover} $i $oldIndex]
set fvalue [$func $newCoords]
incr evals
#puts "$newCoords -- $fvalue -- $evals"
if { $fvalue < [lindex $point 1] } {
set renewed [expr {$i == $oldIndex? 1 : 0}] ;# Is the best estimate being replaced?
set newPoint [list $newCoords $fvalue]
} else {
set newPoint $point
}
#puts "$newPoint -- $evals"
lappend newPoints $newPoint
}
#
# Check the number of evaluations ... not quite accurate, but it will do
#
# Hm, this will fail if this happens in the first generation
#
if { $evals >= ${-maxevaluations} } {
#puts "Maximum evaluations reached"
break
}
#
# Get the best point in the current generation
#
set bestIndex [IndexBestPoint $newPoints]
set bestValue [lindex $newPoints $bestIndex 1]
#puts "$oldIndex -- $oldValue -- $bestIndex -- $bestValue"
#if { $renewed } {}
if { ( $oldValue != $bestValue || $oldIndex != $bestIndex ) &&
( ($oldValue - $bestValue) <= ${-abstolerance} ||
($oldValue - $bestValue) <= 0.5 * ${-reltolerance} * (abs($oldValue) + abs($bestValue)) ) } {
#puts "Values: $oldValue -- $bestValue"
break
} else {
set points $newPoints
set oldIndex $bestIndex
set oldValue $bestValue
lappend bestPerGeneration $bestValue
}
#puts "$oldIndex -- $oldValue -- $bestIndex -- $bestValue"
}
return [dict create optimum-coordinates [lindex $newPoints $bestIndex 0] \
optimum-value [lindex $newPoints $bestIndex 1] evaluations $evals best-values $bestPerGeneration]
}
# ConstructNewCoords --
# Constructs the coordinates of a new point using the DE method
#
# Arguments:
# points Current set of points (each together with the function value)
# factor Weight for the difference vector
# lambda Weight for the best vector
# crossover Probability of cross-over
# idx Current index
# bestIdx Index of the current best vector
#
# Result:
# List of coordinates
#
proc ::math::probopt::ConstructNewCoords {points factor lambda crossover idx bestIdx} {
set number [llength $points]
set dims [llength [lindex $points 0 0]]
set r1 [SelectIndex $idx $number]
set r2 [SelectIndex $idx $number]
set r3 [SelectIndex $idx $number]
if { $lambda == 0.0 } {
set vcoords {}
foreach c1 [lindex $points $r1 0] \
c2 [lindex $points $r2 0] \
c3 [lindex $points $r3 0] {
set vc [expr {$c1 + $factor * ($c2 - $c3)}]
lappend vcoords $vc
}
} else {
set vcoords {}
foreach c1 [lindex $points $idx 0] \
cb [lindex $points $bestIndex 0] \
c2 [lindex $points $r2 0] \
c3 [lindex $points $r3 0] {
set vc [expr {$c1 + $lambda * ($cb - $c1) + $factor * ($c2 - $c3)}]
lappend vcoords $vc
}
}
#
# Now the cross-over per dimension
#
set start [SelectIndex {} $number]
set length [SelectLength $crossover $dims]
set combined $vcoords
for {set i $start} {$i < $start+$length} {incr i} {
set j [expr {$i % $dims}]
lset combined $j [lindex $vcoords $j]
}
return $combined
}
# SelectIndex --
# Select a random index unequal to a given index
#
# Arguments:
# avoidIdx Index to be avoided
# maximum Maximum + 1 for the index
#
# Result:
# Random index in [0,maximum-1], not equal avoidIdx
#
proc ::math::probopt::SelectIndex {avoidIdx maximum} {
set idx $avoidIdx
while { $idx == $avoidIdx } {
set idx [expr {int($maximum * rand())}]
}
return $idx
}
# SelectLength --
# Select a random length using a cross-over probability
#
# Arguments:
# crossover Cross-over probability
# maximum Maximum + 1 for the index
#
# Result:
# Random index in [0,maximum-1]
#
proc ::math::probopt::SelectLength {crossover maximum} {
set length 0
while {1} {
incr length
if { rand() > $crossover || $length >= $maximum } {
break
}
}
return $length
}
# GeneratePoint --
# Generate the coordinates of a random point within the given bounds
#
# Arguments:
# bounds Bounds on all coordinates
#
# Result:
# List of coordinates
#
proc ::math::probopt::GeneratePoint {bounds} {
set coords {}
foreach bound $bounds {
lassign $bound cmin cmax
lappend coords [expr {$cmin + ($cmax - $cmin) * rand()}]
}
return $coords
}
# IndexBestPoint --
# Find the index of the best point (lowest function value)
#
# Arguments:
# points List of points (each is a pair of coordinates and the function value)
#
# Result:
# Index of the best point
#
proc ::math::probopt::IndexBestPoint {points} {
set index 0
set bestValue [lindex $points 0 1]
for {set i 1} {$i < [llength $points]} {incr i} {
set newValue [lindex $points $i 1]
if { $newValue < $bestValue } {
set index $i
set bestValue $newValue
}
}
return $index
}
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