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# -*- tcl -*-
# Tests for 1-d optimisation functions in math library -*- tcl -*-
#
# This file contains a collection of tests for one or more of the Tcllib
# procedures. Sourcing this file into Tcl runs the tests and
# generates output for errors. No output means no errors were found.
#
# $Id: optimize.test,v 1.17 2011/01/18 07:49:53 arjenmarkus Exp $
#
# Copyright (c) 2004 by Arjen Markus
# Copyright (c) 2004, 2005 by Kevin B. Kenny
# All rights reserved.
#
# Note:
# By evaluating the tests in a different namespace than global,
# we assure that the namespace issue (Bug #...) is checked.
#
# -------------------------------------------------------------------------
source [file join \
[file dirname [file dirname [file join [pwd] [info script]]]] \
devtools testutilities.tcl]
testsNeedTcl 8.5
testsNeedTcltest 2.1
support {
useLocal math.tcl math
}
testing {
useLocal optimize.tcl math::optimize
}
# -------------------------------------------------------------------------
namespace eval optimizetest {
namespace import ::math::optimize::*
set old_precision $::tcl_precision
if {![package vsatisfies [package present Tcl] 8.5]} {
set ::tcl_precision 17
} else {
set ::tcl_precision 0
}
#
# Simple test functions
#
proc const_func { x } {
return 1.0
}
proc ffunc { x } {
expr {$x*(1.0-$x*$x)}
}
proc minfunc { x } {
expr {-$x*(1.0-$x*$x)}
}
proc absfunc { x } {
expr {abs($x*(1.0-$x*$x))}
}
proc within_range { result min max } {
#puts "Within range? $result $min $max"
#puts "[expr {2.0*abs($result-$min)/abs($max+$min)}]"
if { $result >= $min && $result <= $max } {
set ok 1
} else {
set ok 0
}
return $ok
}
#
# Test the minimum procedure
#
# Note about the uneven and even functions:
# the initial interval is chosen symmetrical, so that the
# three function values are equal.
#
test optimize-1.1 "Minimum of constant function" {
set result [minimum -1.0 1.0 ::optimizetest::const_func]
within_range $result -1.0 1.0
} 1
test optimize-1.2 "Minimum of odd function, case 1" {
set result [minimum -1.0 1.0 ::optimizetest::ffunc]
set xmin [expr {-sqrt(1.0/3.0)-0.0001}]
set xmax [expr {-sqrt(1.0/3.0)+0.0001}]
within_range $result $xmin $xmax
} 1
test optimize-1.3 "Minimum of odd function, asymmetric interval" {
set result [minimum -0.8 1.2 ::optimizetest::ffunc]
set xmin [expr {-sqrt(1.0/3.0)-0.0001}]
set xmax [expr {-sqrt(1.0/3.0)+0.0001}]
within_range $result $xmin $xmax
} 1
test optimize-1.4 "Minimum of odd function, case 2" {
set result [minimum -1.0 1.0 ::optimizetest::minfunc]
set xmin [expr {sqrt(1.0/3.0)-0.0001}]
set xmax [expr {sqrt(1.0/3.0)+0.0001}]
within_range $result $xmin $xmax
} 1
test optimize-1.5 "Minimum of even function" {
set result [minimum -1.0 1.0 ::optimizetest::absfunc]
set xmin -0.0001
set xmax 0.0001
within_range $result $xmin $xmax
} 1
#
# Test the maximum procedure
#
# Note about the uneven and even functions:
# the initial interval is chosen symmetrical, so that the
# three function values are equal.
#
test optimize-2.1 "Maximum of constant function" {
set result [maximum -1.0 1.0 ::optimizetest::const_func]
within_range $result -1.0 1.0
} 1
test optimize-2.2 "Maximum of odd function, case 1" {
set result [maximum -1.0 1.0 ::optimizetest::ffunc]
set xmin [expr {sqrt(1.0/3.0)-0.0001}]
set xmax [expr {sqrt(1.0/3.0)+0.0001}]
within_range $result $xmin $xmax
} 1
test optimize-2.3 "Maximum of odd function, case 2" {
set result [maximum -1.0 1.0 ::optimizetest::minfunc]
set xmin [expr {-sqrt(1.0/3.0)-0.0001}]
set xmax [expr {-sqrt(1.0/3.0)+0.0001}]
within_range $result $xmin $xmax
} 1
#
# Either of the two maxima will do
#
test optimize-2.4 "Maximum of even function" {
set result [maximum -1.0 1.0 ::optimizetest::absfunc]
set xmin [expr {-sqrt(1.0/3.0)-0.0001}]
set xmax [expr {-sqrt(1.0/3.0)+0.0001}]
set ok [within_range $result $xmin $xmax]
set xmin [expr {sqrt(1.0/3.0)-0.0001}]
set xmax [expr {sqrt(1.0/3.0)+0.0001}]
incr ok [within_range $result $xmin $xmax]
} 1
# Custom match procedure for approximate results
proc withinEpsilon { shouldBe is } {
expr { [string is double $is]
&& abs( $is - $shouldBe ) < 1.e-07 * abs($shouldBe) }
}
::tcltest::customMatch withinEpsilon [namespace code withinEpsilon]
test linmin-1.1 {find minimum of a parabola - constrained} \
-setup {
proc f x { expr { ($x + 3.) * ($x - 1.) } }
} \
-body {
foreach {x y} [min_bound_1d f 10. -10.] break
set x
} \
-cleanup {
rename f {}
} \
-result -1. \
-match withinEpsilon
test linmin-1.2 {find minimum of cosine} \
-setup {
proc f x { expr { cos($x) } }
} \
-body {
foreach { x y } [min_bound_1d f 0. 6.28318] break
set x
} \
-cleanup {
rename f {}
} \
-result 3.1415926535897932 \
-match withinEpsilon
test linmin-1.3 {find minimum of a bell-shaped function} \
-setup {
proc f x {
set t [expr { $x - 3. }]
return [expr { -exp ( -$t * $t / 2 ) }]
}
} \
-body {
foreach { x y } [min_bound_1d f 0 30.] break
set x
} \
-cleanup {
rename f {}
} \
-result 3. \
-match withinEpsilon
test linmin-1.4 {function where parabolic extrapolation never works} \
-setup {
proc f x { expr { -1. / ( 0.01 + abs( $x - 5.) ) } }
} \
-body {
foreach {x y} [min_bound_1d f 0 20.] break
set x
} \
-cleanup {
rename f {}
} \
-result 5. \
-match withinEpsilon
test linmin-2.1 {wrong \# args} \
-body {
min_bound_1d f
} \
-returnCodes 1 \
-result [tcltest::wrongNumArgs min_bound_1d {f x1 x2 args} 1]
test linmin-2.2 {wrong \# args} \
-body {
min_bound_1d f 0 1 -bad
} \
-returnCodes 1 \
-result "wrong # args, should be \"min_bound_1d f x1 x2 ?-option value?...\""
test linmin-2.3 {bad arg} \
-body {
min_bound_1d f 0 1 -bad option
} \
-returnCodes 1 \
-result "unknown option \"-bad\", should be -abserror,\
-fguess, -guess, -initial,\
-maxiter, -relerror, or -trace"
test linmin-2.4 {iteration limit} \
-setup {
proc f x { expr { -1. / ( 0.01 + abs( $x - 5.) ) } }
} \
-body {
min_bound_1d f 20. 0 -maxiter 10
} \
-cleanup {
rename f {}
} \
-returnCodes 1 \
-result "min_bound_1d failed to converge after \\d* steps" \
-match regexp
test linmin-3.1 {minimise cos(x), unbounded} \
-setup {
proc f x { expr { cos($x) } }
} -body {
foreach { x y } [min_unbound_1d f 3. 3.01] break
set x
} \
-cleanup {
rename f {}
} \
-result 3.1415926535897932 \
-match withinEpsilon
test linmin-3.2 {minimise cos(x), unbounded, too eager} \
-setup {
proc f x { expr { cos($x) } }
} -body {
foreach { x y } [min_unbound_1d f 0.1 0.15] break
set x
} \
-cleanup {
rename f {}
} \
-result [expr { 3. * 3.1415926535897932 }] \
-match withinEpsilon
test linmin-3.3 {near underflow in parabolic extrapolation} \
-setup {
proc f x {
expr { ( 1.12712e-22 * $x * $x * $x - 1e-15 ) * $x + 1e-15 }
}
} \
-body {
foreach { x y } [min_unbound_1d f 1. 0.] break
set x
} \
-cleanup {
rename f {}
} \
-result 130.41372 \
-match withinEpsilon
test linmin-3.4 {near underflow in parabolic extrapolation} \
-setup {
proc f x {
expr { ( ( 1e-30 * $x * $x - 1.12712e-22 )
* $x * $x * $x - 1e-15 )
* $x + 1e-15 }
}
} \
-body {
foreach { x y } [min_unbound_1d f 1. 0. -relerror 1e-08] break
set x
} \
-cleanup {
rename f {}
} \
-result 8668.4248 \
-match withinEpsilon
test linmin-3.5 {parabolic interpolation finds a minimum - case 1} \
-setup {
proc f x {
expr { ( ( ( 1e-5 * $x - 2.69672 )
* $x + 10.0902 )
* $x - 8.39345 )
* $x + 1. }
}
} \
-body {
foreach { x y } [min_unbound_1d f 1. 0. -relerror 1e-08] break
set x
} \
-cleanup {
rename f {}
} \
-result 0.527450252 \
-match withinEpsilon
test linmin-3.6 {parabolic interpolation finds a minimum - case 2} \
-setup {
proc f x {
expr { ( ( 0.125669 * $x * $x - 0.982687 )
* $x - 0.142982 )
* $x + 1 }
}
} \
-body {
foreach { x y } [min_unbound_1d f 1. 0. -relerror 1e-08] break
set x
} \
-cleanup {
rename f {}
} \
-result 2.0127451 \
-match withinEpsilon
test linmin-3.7 {parabolic interpolation is useless} \
-setup {
proc f x {
expr { ( ( ( 1e-5 * $x - 6.79171 )
* $x + 24.8107 )
* $x - 19.019 )
* $x + 1. }
}
} \
-body {
foreach { x y } [min_unbound_1d f 1 0 -relerror 1e-8] break
set x
} \
-cleanup {
rename f {}
} \
-result 509375.81 \
-match withinEpsilon
test linmin-4.1 {wrong \# args} \
-body {
min_unbound_1d f
} \
-returnCodes 1 \
-result [tcltest::wrongNumArgs min_unbound_1d {f x1 x2 args} 1]
test linmin-4.2 {wrong \# args} \
-body {
min_unbound_1d f 0 1 -bad
} \
-returnCodes 1 \
-result "wrong # args, should be \"min_unbound_1d f x1 x2 ?-option value?...\""
test linmin-4.3 {bad arg} \
-body {
min_unbound_1d f 0 1 -bad option
} \
-returnCodes 1 \
-result "unknown option \"-bad\", should be -trace"
#
# Test the solveLinearProgram procedure
#
set ::symm_constraints {
{ 1.0 2.0 1.0 }
{ 2.0 1.0 1.0 } }
test linprog-1.0 "Symmetric constraints, case 1" \
-body {
set result [solveLinearProgram {1.0 1.0} $::symm_constraints]
set ok 1
if { ! [within_range [lindex $result 0] 0.333300 0.333360] ||
! [within_range [lindex $result 1] 0.333300 0.333360] } {
set ok 0
}
set ok
} \
-result 1
test linprog-1.1 "Symmetric constraints, case 2" \
-body {
set result [solveLinearProgram {1.0 0.0} $::symm_constraints]
set ok 1
if { ! [within_range [lindex $result 0] 0.49900 0.50100] ||
! [within_range [lindex $result 1] -0.00100 0.00100] } {
set ok 0
}
set ok
} \
-result 1
test linprog-1.2 "Symmetric constraints, case 3" \
-body {
set result [solveLinearProgram {0.0 1.0} $::symm_constraints]
set ok 1
if { ! [within_range [lindex $result 1] 0.499900 0.500100] ||
! [within_range [lindex $result 0] -0.000100 0.000100] } {
set ok 0
}
set ok
} \
-result 1
test linprog-1.3 "Symmetric constraints, case 4" \
-body {
set result [solveLinearProgram {3.0 4.0} $::symm_constraints]
set ok 1
if { ! [within_range [lindex $result 0] 0.333300 0.333360] ||
! [within_range [lindex $result 1] 0.333300 0.333360] } {
set ok 0
}
set ok
} \
-result 1
test linprog-2.1 "Unbounded program 1" \
-body {
set result [solveLinearProgram {3.0 4.0} {{1.0 -2.0 1.0} {-2.0 1.0 1.0}} ]
} \
-result "unbounded"
test linprog-2.2 "Unbounded program 2" \
-body {
set result [::math::optimize::solveLinearProgram {2.0 1.0} {{3.0 0.0 6.0} {1.0 0.0 2.0}}]
} \
-result "unbounded"
test linprog-2.3 "Infeasible program" \
-body {
set result [::math::optimize::solveLinearProgram {2.0 1.0} {{3.0 1.0 6.0} {1.0 -1.0 2.0} {0.0 1.0 -3.0}}]
} \
-result "infeasible"
test linprog-2.4 "Degenerate program" \
-body {
# Solution: {1.0 3.0}
set result [::math::optimize::solveLinearProgram {2.0 1.0} {{3.0 1.0 6.0} {1.0 -1.0 2.0} {0.0 1.0 3.0}}]
set ok 1
if { ! [within_range [lindex $result 0] 0.99999 1.00001] ||
! [within_range [lindex $result 1] 2.99999 3.00001] } {
set ok 0
}
set ok
} \
-result 1
test linprog-3.1 "Simple 3D program" \
-body {
set result [solveLinearProgram \
{1.0 1.0 1.0} \
{{1.0 1.0 2.0 1.0}
{1.0 2.0 1.0 1.0}
{2.0 1.0 1.0 1.0}}]
set ok 1
if { ! [within_range [lindex $result 0] 0.249900 0.250100] ||
! [within_range [lindex $result 1] 0.249900 0.250100] ||
! [within_range [lindex $result 2] 0.249900 0.250100] } {
set ok 0
}
set ok
} \
-result 1
test nelderMead-1.1 "Nelder-Mead - wrong \# args" \
-body {
::math::optimize::nelderMead f {0.0 0.0} -bogus
} \
-returnCodes error \
-match glob \
-result "wrong \# args*"
test nelderMead-1.2 "Nelder-Mead - bad param" \
-body {
::math::optimize::nelderMead f {0.0 0.0} -bogus 1
} \
-returnCodes error \
-match glob \
-result {unknown option "-bogus"*}
test nelderMead-1.3 "Nelder-Mead - bad size of scale" \
-body {
::math::optimize::nelderMead f {0.0 0.0} -scale {0 0 0}
} \
-returnCodes error \
-result {-scale vector must be of same size as starting x vector}
# Easy case - minimize in a paraboloid
test nelderMead-2.1 "Nelder-Mead - easy" \
-setup {
proc f {x y} {
expr {($x-3.)*($x-3.) + ($y-2.)*($y-2.) + 1.}
}
} \
-body {
array set dd [::math::optimize::nelderMead f {1. 1.}]
foreach {x y} $dd(x) break
expr { abs($x-3.) < 0.001 && abs($y-2.) < 0.001 }
} \
-cleanup {
rename f {}; unset dd
} \
-result 1
test nelderMead-2.2 "Nelder-Mead - easy" \
-setup {
proc f {x y} {
expr {($x-3.)*($x-3.) + ($y-2.)*($y-2.) + 1.}
}
} \
-body {
array set dd [::math::optimize::nelderMead f {0. 0.}]
foreach {x y} $dd(x) break
expr { abs($x-3.) < 0.001 && abs($y-2.) < 0.001 }
} \
-cleanup {
rename f {}; unset dd
} \
-result 1
# Slalom down a sinuous valley - exercises most of the code
test nelderMead-2.3 "Nelder-Mead - sinuous valley" \
-setup {
set pi 3.1415926535897932
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
}
} \
-body {
array set dd [::math::optimize::nelderMead f {1. 0.} -scale {0.1 0.01}]
foreach {x y} $dd(x) break
expr { abs($x-$pi/2) < 0.001 && abs($y-0.1) < 0.001 }
} \
-cleanup {rename f {}; unset dd} \
-result 1
# Exercise the difficult case where the simplex has to contract about the
# low point because all else has failed.
test nelderMead-2.4 "Nelder-Mead - simplex contracts about the minimum" \
-setup {
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
}
} \
-body {
array set dd [::math::optimize::nelderMead g {0. 0.} \
-scale {1. 1.} -ftol 1e-10]
foreach {x y} $dd(x) break
expr { abs($x-0.774561) < 0.00005 && abs($y-0.755644) < 0.00005 }
} \
-cleanup {
rename g {}; unset dd
} \
-result 1
# Make sure the method deals gracefully with a "valley"
# (Ticket UUID: 3193459)
test nelderMead-2.5 "Nelder-Mead - indeterminate minimum (valley)" \
-setup {
proc h {a b} {
return [expr {abs($a-$b)}]
}
} \
-body {
array set dd [::math::optimize::nelderMead h {1. 1.}]
foreach {x y} $dd(x) break
expr { abs($x-1.) < 0.00005 && abs($y-1.) < 0.00005 }
} \
-cleanup {
rename h {}; unset dd
} \
-result 1
testsuiteCleanup
# Restore precision
set ::tcl_precision $old_precision
# Local Variables:
# mode: tcl
# End:
} ;# End of optimizetest namespace
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