#! perl
sub sqr ( $x ) {
return $x * $x;
}
PI2 => 6.283185307179586,
PI3 => 9.424777960769379,
PI4 => 12.5663706143592,
};
sub rosenbrock_f ( $n, $x, $grad, $data ) {
my \@x = $x;
my $a = $x[1] - $x[0] * $x[0];
my $b = 1 - $x[0];
if ( $grad ) {
my \@grad = $grad;
$grad[0] = -400 * $a * $x[0] - 2 * $b;
$grad[1] = 200 * $a;
}
return 100 * sqr( $a ) + sqr( $b );
}
my @rosenbrock_lb = ( -2, -2 );
my @rosenbrock_ub = ( 2, 2 );
my @rosenbrock_xmin = ( 1, 1 );
sub mccormic_f ( $n, $x, $grad, $data ) {
my \@x = $x;
my $a = $x[0] + $x[1];
my $b = $x[0] - $x[1];
if ( $grad ) {
my \@grad = $grad;
$grad[0] = cos( $a ) + 2 * $b - 1.5;
$grad[1] = cos( $a ) - 2 * $b + 2.5;
}
return sin( $a ) + sqr( $b ) - 1.5 * $x[0] + 2.5 * $x[1] + 1;
}
my @mccormic_lb = ( -1.5, -3 );
my @mccormic_ub = ( 4, 4 );
my @mccormic_xmin = ( -0.547197553, -1.54719756 );
sub boxbetts_f ( $n, $x, $grad, $data ) {
my $i;
my $f = 0;
my \@x = $x;
if ( $grad ) {
$grad->@* = ( 0 ) x $n;
}
for my $i ( 1 .. 10 ) {
my $e0 = exp( -0.1 * $i * $x[0] );
my $e1 = exp( -0.1 * $i * $x[1] );
my $e2 = exp( -0.1 * $i ) - exp( -1.0 * $i );
my $g = $e0 - $e1 - $e2 * $x[2];
$f += sqr( $g );
if ( $grad ) {
my \@grad = $grad;
$grad[0] += ( 2 * $g ) * ( -0.1 * $i * $e0 );
$grad[1] += ( 2 * $g ) * ( 0.1 * $i * $e1 );
$grad[2] += -( 2 * $g ) * $e2;
}
}
return $f;
}
my @boxbetts_lb = ( 0.9, 9, 0.9 );
my @boxbetts_ub = ( 1.2, 11.2, 1.2 );
my @boxbetts_xmin = ( 1, 10, 1 );
sub paviani_f ( $n, $x, $grad, $data ) {
my \@x = 0;
my $f = 0;
my $prod = 1;
if ( $grad ) {
$grad->@* = ( 0 ) x $n;
}
for my $i ( 0 .. 9 ) {
my $ln1 = log( $x[$i] - 2 );
my $ln2 = log( 10 - $x[$i] );
$f += sqr( $ln1 ) + sqr( $ln2 );
if ( $grad ) {
$grad->[$i] += 2 * $ln1 / ( $x[$i] - 2 ) - 2 * $ln2 / ( 10 - $x[$i] );
}
$prod *= $x[$i];
}
$f -= ( $prod = pow( $prod, 0.2 ) );
if ( $grad ) {
$grad->[$_] -= 0.2 * $prod / $x[$_] for 0 .. 9;
}
return ( $f );
}
my @paviani_lb = ( 2.001, 2.001, 2.001, 2.001, 2.001, 2.001, 2.001, 2.001, 2.001, 2.001 );
my @paviani_ub = ( 9.999, 9.999, 9.999, 9.999, 9.999, 9.999, 9.999, 9.999, 9.999, 9.999 );
my @paviani_xmin = (
9.35026583, 9.35026583, 9.35026583, 9.35026583, 9.35026583, 9.35026583,
9.35026583, 9.35026583, 9.35026583, 9.35026583
);
sub grosenbrock_f ( $n, $x, $grad, $data ) {
my \@x = $x;
my $f = 0;
if ( $grad ) {
$grad->[0] = 0;
}
for my $i ( 0 .. 28 ) {
my $a = $x[ $i + 1 ] - $x[$i] * $x[$i];
my $b = 1 - $x[$i];
if ( $grad ) {
$grad->[$i] += -400 * $a * $x[$i] - 2 * $b;
$grad->[ $i + 1 ] = 200 * $a;
}
$f += 100 * sqr( $a ) + sqr( $b );
}
return ( $f );
}
my @grosenbrock_lb = (
-30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30,
-30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30
);
my @grosenbrock_ub = (
30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30,
30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30
);
my @grosenbrock_xmin
= ( 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 );
sub goldsteinprice_f ( $n, $x, $grad, $data ) {
my \@x = $x;
my $x0 = $x[0];
my $x1 = $x[1];
my $a1 = $x0 + $x1 + 1;
my $a12 = sqr( $a1 );
my $a2 = 19 - 14 * $x0 + 3 * $x0 * $x0 - 14 * $x1 + 6 * $x0 * $x1 + 3 * $x1 * $x1;
my $b1 = 2 * $x0 - 3 * $x1;
my $b12 = sqr( $b1 );
my $b2 = 18 - 32 * $x0 + 12 * $x0 * $x0 + 48 * $x1 - 36 * $x0 * $x1 + 27 * $x1 * $x1;
if ( $grad ) {
my \@grad = $grad;
$grad[0] = ( 1 + $a12 * $a2 ) * ( 2 * $b1 * 2 * $b2 + $b12 * ( -32 + 24 * $x0 - 36 * $x1 ) )
+ ( 2 * $a1 * $a2 + $a12 * ( -14 + 6 * $x0 + 6 * $x1 ) ) * ( 30 + $b12 * $b2 );
$grad[1] = ( 1 + $a12 * $a2 ) * ( 2 * $b1 * ( -3 ) * $b2 + $b12 * ( 48 - 36 * $x0 + 54 * $x1 ) )
+ ( 2 * $a1 * $a2 + $a12 * ( -14 + 6 * $x0 + 6 * $x1 ) ) * ( 30 + $b12 * $b2 );
}
return ( ( 1 + $a12 * $a2 ) * ( 30 + $b12 * $b2 ) );
}
my @goldsteinprice_lb = ( -2, -2 );
my @goldsteinprice_ub = ( 2, 2 );
my @goldsteinprice_xmin = ( 0, -1 );
sub shekel_f ( $n, $x, $grad, $data ) {
my \@x = $x;
my @A = (
[ 4, 4, 4, 4 ],
[ 1, 1, 1, 1 ],
[ 8, 8, 8, 8 ],
[ 6, 6, 6, 6 ],
[ 3, 7, 3, 7 ],
[ 2, 9, 2, 9 ],
[ 5, 5, 3, 3 ],
[ 8, 1, 8, 1 ],
[ 6, 2, 6, 2 ],
[ 7, 3.6, 7, 3.6 ],
);
my @c = ( .1, .2, .2, .4, .4, .6, .3, .7, .5, .5 );
my $f = 0;
my $m = $data;
if ( $grad ) {
$grad->@* = ( 0 ) x $n;
}
for my $i ( 0 .. $m - 1 ) {
my $fi
= 1.0 / ( $c[$i]
+ sqr( $x[0] - $A[$i][0] )
+ sqr( $x[1] - $A[$i][1] )
+ sqr( $x[2] - $A[$i][2] )
+ sqr( $x[3] - $A[$i][3] ) );
$f -= $fi;
if ( $grad ) {
my \@grad = $grad;
$grad[0] += ( 2 * $fi * $fi ) * ( $x[0] - $A[$i][0] );
$grad[1] += ( 2 * $fi * $fi ) * ( $x[1] - $A[$i][1] );
$grad[2] += ( 2 * $fi * $fi ) * ( $x[2] - $A[$i][2] );
$grad[3] += ( 2 * $fi * $fi ) * ( $x[3] - $A[$i][3] );
}
}
return ( $f );
}
my @shekel_m = ( 5, 7, 10 );
my @shekel_lb = ( 0, 0, 0, 0 );
my @shekel_ub = ( 10, 10, 10, 10 );
my @shekel0_xmin = ( 4.000037154, 4.000133276, 4.000037154, 4.000133276 );
my @shekel1_xmin = ( 4.000572917, 4.000689366, 3.999489709, 3.999606158 );
my @shekel2_xmin = ( 4.000746531, 4.000592935, 3.999663399, 3.999509801 );
sub levy_f ( $n, $x, $grad, $data ) {
my \@x = $x;
unsigned i;
my $a = $x[ $n - 1 ] - 1;
my $b = 1 + sqr( sin( PI2 * $x[ $n - 1 ] ) );
my $f = sqr( sin( PI3 * $x[0] ) ) + $a * $b;
if ( $grad ) {
my \@grad = $grad;
@grad = 0;
$grad[0] = 2 * PI3 * sin( PI3 * $x[0] ) * cos( PI3 * $x[0] );
$grad[ $n - 1 ] += $b + $a * 2 * PI2 * sin( PI2 * $x[ $n - 1 ] ) * cos( PI2 * $x[ $n - 1 ] );
}
for my $i ( 0 .. $n - 1 ) {
$a = $x[$i] - 1;
$b = 1 + sqr( sin( PI3 * $x[ $i + 1 ] ) );
$f += sqr( $a ) * $b;
if ( $grad ) {
my \@grad = $grad;
$grad[$i] += 2 * $a * $b;
$grad[ $i + 1 ] += 2 * PI3 * sqr( $a ) * sin( PI3 * $x[ $i + 1 ] ) * cos( PI3 * $x[ $i + 1 ] );
}
}
return ( $f );
}
my @levy_lb = ( -5, -5, -5, -5, -5, -5, -5 );
my @levy_ub = ( 5, 5, 5, 5, 5, 5, 5 );
my @levy_xmin = ( 1, 1, 1, 1, 1, 1, -4.75440246 );
my @levy4_lb = ( -10, -10, -10, -10 );
my @levy4_ub = ( 10, 10, 10, 10 );
my @levy4_xmin = ( 1, 1, 1, -9.75235596 );
sub griewank_f ( $n, $x, $grad, $data ) {
my \@x = $x;
unsigned i;
my $f = 1;
my $p = 1;
for my $i ( 0 .. $n - 1 ) {
$f += sqr( $x[$i] ) * 0.00025;
$p *= cos( $x[$i] / sqrt( $i + 1. ) );
if ( $grad ) {
$grad->[$i] = $x[$i] * 0.0005;
}
}
$f -= $p;
if ( $grad ) {
for my $i ( 0 .. $n - 1 ) {
$grad->[$i] += $p * tan( $x[$i] / sqrt( $i + 1. ) ) / sqrt( $i + 1. );
}
}
return ( $f );
}
my @griewank_lb = ( -500, -500, -500, -500, -500, -500, -500, -500, -500, -500 );
my @griewank_ub = ( 600, 600, 600, 600, 600, 600, 600, 600, 600, 600 );
my @griewank_xmin = ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 );
sub sixhumpcamel_f ( $n, $x, $grad, $data ) {
my \@x = $x;
if ( $grad ) {
my \@grad = $grad;
$grad[0] = 8 * $x[0] - 2.1 * 4 * pow( $x[0], 3. ) + 2 * pow( $x[0], 5. ) + $x[1];
$grad[1] = $x[0] - 8 * $x[1] + 16 * pow( $x[1], 3. );
}
return (4 * sqr( $x[0] )
- 2.1 * pow( $x[0], 4. )
+ pow( $x[0], 6. ) / 3.
+ $x[0] * $x[1]
- 4 * sqr( $x[1] )
+ 4 * pow( $x[1], 4. ) );
}
my @sixhumpcamel_lb = ( -5, -5 );
my @sixhumpcamel_ub = ( 5, 5 );
my @sixhumpcamel_xmin = ( 0.08984201317, -0.7126564032 );
sub convexcosh_f ( $n, $x, $grad, $data ) {
my \@x = $x;
my $f = 1;
for my $i ( 0 .. $n - 1 ) {
$f *= cosh( ( $x[$i] - $i ) * ( $i + 1 ) );
}
if ( $grad ) {
my \@grad = $grad;
for my $i ( 0 .. $n - 1 ) {
$grad[$i] = $f * tanh( ( $x[$i] - $i ) * ( $i + 1 ) ) * ( $i + 1 );
}
}
return ( $f );
}
my @convexcosh_lb = ( -1, 0, 0, 0, 0, 0, 0, 0, 0, 0 );
my @convexcosh_ub = ( 2, 3, 6, 7, 8, 10, 11, 13, 14, 16 );
my @convexcosh_xmin = ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 );
sub branin_f ( $n, $x, $grad, $data ) {
my \@x = $x;
my $a = 1 - 2 * $x[1] + 0.05 * sin( PI4 * $x[1] ) - $x[0];
my $b = $x[1] - 0.5 * sin( PI2 * $x[0] );
if ( $grad ) {
my \@grad = $grad;
$grad[0] = -2 * $a - cos( PI2 * $x[0] ) * PI2 * $b;
$grad[1] = 2 * $a * ( 0.05 * PI4 * cos( PI4 * $x[1] ) - 2 ) + 2 * $b;
}
return ( sqr( $a ) + sqr( $b ) );
}
my @branin_lb = ( -10, -10 );
my @branin_ub = ( 10, 10 );
my @branin_xmin = ( 1, 0 );
sub shubert_f ( $n, $x, $grad, $data ) {
my \@x = $x;
my $f = 0;
for my $j ( 1 .. 5 ) {
for my $i ( 0 .. $n - 1 ) {
$f -= $j * sin( ( $j + 1 ) * $x[$i] + $j );
}
}
if ( $grad ) {
my \@grad = $grad;
for my $i ( 0 .. $n - 1 ) {
$grad[$i] = 0;
for my $j ( 1 .. 5 ) {
$grad[$i] -= $j * ( $j + 1 ) * cos( ( $j + 1 ) * $x[$i] + $j );
}
}
}
return ( $f );
}
my @shubert_lb = ( -10, -10 );
my @shubert_ub = ( 10, 10 );
my @shubert_xmin = ( -6.774576143, -6.774576143 );
sub hansen_f ( $n, $x, $grad, $data ) {
my \@x = $x;
unsigned i;
my $a = 0;
my $b = 0;
for my $i ( 1 .. 5 ) {
$a += $i * cos( ( $i - 1 ) * $x[0] + $i );
}
for my $i ( 1 .. 5 ) {
$b += $i * cos( ( $i + 1 ) * $x[1] + $i );
}
if ( $grad ) {
my \@grad = $grad;
$grad[0] = 0;
for my $i ( 1 .. 5 ) {
$grad[0] -= $i * ( $i - $1 ) * sin( ( $i - 1 ) * $x[0] + $i );
}
$grad[0] *= $b;
$grad[1] = 0;
for my $i ( 1 .. 5 ) {
$grad[1] -= $i * ( $i + 1 ) * sin( ( $i + 1 ) * $x[1] + $i );
}
$grad[1] *= $a;
}
return ( $a * $b );
}
my @hansen_lb = ( -10, -10 );
my @hansen_ub = ( 10, 10 );
my @hansen_xmin = ( -1.306707704, -1.425128429 );
sub osc1d_f ( $n, $x, $grad, $data ) {
my \@x = $x;
my $y = $x[0] - 1.23456789;
if ( $grad ) {
my \@grad = $grad;
$grad[0] = $y * 0.02 + sin( $y - 2 * sin( 3 * $y ) ) * ( 1 - 6 * cos( 3 * $y ) );
}
return ( sqr( $y * 0.1 ) - cos( $y - 2 * sin( 3 * $y ) ) );
}
my @osc1d_lb = ( -5 );
my @osc1d_ub = ( 5 );
my @osc1d_xmin = ( 1.23456789 );
sub corner4d_f ( $n, $x, $grad, $data ) {
my \@x = $x;
my $u = $x[0] + $x[1] * $x[2] * sin( 2 * $x[3] );
my $v = $x[0] + 2 * sin( $u );
if ( $grad ) {
my \@grad = $grad;
$grad[0] = 2 * $v * ( 1 + 2 * cos( $u ) );
$grad[1] = 2 * $v * 2 * cos( $u ) * $x[2] * sin( 2 * $x[3] ) + 0.1;
$grad[2] = 2 * $v * 2 * cos( $u ) * $x[1] * sin( 2 * $x[3] ) + 0.1;
$grad[3] = 2 * $v * 2 * cos( $u ) * $x[1] * $x[2] * cos( 2 * $x[3] ) * 2 + 0.1;
}
return ( 1 + $v * $v + 0.1 * ( $x[1] + $x[2] + $x[3] ) );
}
my @corner4d_lb = ( 0, 0, 0, 0 );
my @corner4d_ub = ( 1, 1, 1, 1 );
my @corner4d_xmin = ( 0, 0, 0, 0 );
sub side4d_f ( $n, $x, $grad, $data ) {
my \@x = $x;
my $w0 = 0.1;
my $w1 = 0.2;
my $w2 = 0.3;
my $w3 = 0.4;
my $x0 = +0.4977 * $x[0] - 0.3153 * $x[1] - 0.5066 * $x[2] - 0.4391 * $x[3];
my $x1 = -0.3153 * $x[0] + 0.3248 * $x[1] - 0.4382 * $x[2] - 0.4096 * $x[3];
my $x2 = -0.5066 * $x[0] - 0.4382 * $x[1] + 0.3807 * $x[2] - 0.4543 * $x[3];
my $x3 = -0.4391 * $x[0] - 0.4096 * $x[1] - 0.4543 * $x[2] + 0.5667 * $x[3];
my $d0 = -1. / ( $x0 * $x0 + $w0 * $w0 );
my $d1 = -1. / ( $x1 * $x1 + $w1 * $w1 );
my $d2 = -1. / ( $x2 * $x2 + $w2 * $w2 );
my $d3 = -1. / ( $x3 * $x3 + $w3 * $w3 );
if ( $grad ) {
my \@grad = $grad;
$grad[0]
= 2 * ( $x0 * $d0 * $d0 * +0.4977
+ $x1 * $d1 * $d1 * -0.3153
+ $x2 * $d2 * $d2 * -0.5066
+ $x3 * $d3 * $d3 * -0.4391 );
$grad[1]
= 2 * ( $x0 * $d0 * $d0 * -0.3153
+ $x1 * $d1 * $d1 * +0.3248
+ $x2 * $d2 * $d2 * -0.4382
+ $x3 * $d3 * $d3 * -0.4096 );
$grad[2]
= 2 * ( $x0 * $d0 * $d0 * -0.5066
+ $x1 * $d1 * $d1 * -0.4382
+ $x2 * $d2 * $d2 * +0.3807
+ $x3 * $d3 * $d3 * -0.4543 );
$grad[3]
= 2 * ( $x0 * $d0 * $d0 * -0.4391
+ $x1 * $d1 * $d1 * -0.4096
+ $x2 * $d2 * $d2 * -0.4543
+ $x3 * $d3 * $d3 * +0.5667 );
}
return ( $d0 + $d1 + $d2 + $d3 );
}
my @side4d_lb = ( 0.1, -1, -1, -1 );
my @side4d_ubs = ( 1, 1, 1, 1 );
my @side4d_xmin = ( 0.1, 0.102971169, 0.0760520641, -0.0497098571 );
my @testfuncs = (
[
\&rosenbrock_f, undef, 1, 2,
\&rosenbrock_lb, \&rosenbrock_ub, \&rosenbrock_xmin, 0.0,
"Rosenbrock function"
],
[
\&mccormic_f, undef, 1, 2,
\&mccormic_lb, \&mccormic_ub, \&mccormic_xmin, -1.91322295,
"McCormic function"
],
[
\&boxbetts_f, undef, 1, 3, \&boxbetts_lb, \&boxbetts_ub, \&boxbetts_xmin,
0.0, "Box and Betts exponential quadratic sum"
],
[
\&paviani_f, undef, 1, 10,
\&paviani_lb, \&paviani_ub, \&paviani_xmin, -45.7784697,
"Paviani function"
],
[
\&grosenbrock_f, undef, 1, 30, \&grosenbrock_lb, \&grosenbrock_ub, \&grosenbrock_xmin,
0.0, "Generalized Rosenbrock function"
],
[
\&goldsteinprice_f, undef, 1, 2, \&goldsteinprice_lb, \&goldsteinprice_ub, \&goldsteinprice_xmin,
3.0, "Goldstein and Price function"
],
[
\&shekel_f, \&shekel_m +0, 1, 4,
\&shekel_lb, \&shekel_ub, \&shekel0_xmin, -10.15319968,
"Shekel m=5 function"
],
[
\&shekel_f, \&shekel_m + 1, 1, 4,
\&shekel_lb, \&shekel_ub, \&shekel1_xmin, -10.40294057,
"Shekel m=7 function"
],
[
\&shekel_f, \&shekel_m + 2, 1, 4,
\&shekel_lb, \&shekel_ub, \&shekel2_xmin, -10.53640982,
"Shekel m=10 function"
],
[ \&levy_f, undef, 1, 4, \&levy4_lb, \&levy4_ub, \&levy4_xmin, -21.50235596, "Levy n=4 function" ],
[ \&levy_f, undef, 1, 5, \&levy_lb, \&levy_ub, \&levy_xmin + 2, -11.50440302, "Levy n=5 function" ],
[ \&levy_f, undef, 1, 6, \&levy_lb, \&levy_ub, \&levy_xmin + 1, -11.50440302, "Levy n=6 function" ],
[ \&levy_f, undef, 1, 7, \&levy_lb, \&levy_ub, \&levy_xmin, -11.50440302, "Levy n=7 function" ],
[
\&griewank_f, undef, 1, 10,
\&griewank_lb, \&griewank_ub, \&griewank_xmin, 0.0,
"Griewank function"
],
[
\&sixhumpcamel_f, undef, 1, 2, \&sixhumpcamel_lb, \&sixhumpcamel_ub, \&sixhumpcamel_xmin,
-1.031628453, "Six-hump camel back function"
],
[
\&convexcosh_f, undef, 1, 10, \&convexcosh_lb, \&convexcosh_ub, \&convexcosh_xmin,
1.0, "Convex product of cosh functions"
],
[ \&branin_f, undef, 1, 2, \&branin_lb, \&branin_ub, \&branin_xmin, -.0, "Branin function" ],
[
\&shubert_f, undef, 1, 2,
\&shubert_lb, \&shubert_ub, \&shubert_xmin, -24.06249888,
"Shubert function"
],
[
\&hansen_f, undef, 1, 2,
\&hansen_lb, \&hansen_ub, \&hansen_xmin, -176.5417931367,
"Hansen function"
],
[
\&osc1d_f, undef, 1, 1, \&osc1d_lb, \&osc1d_ub, \&osc1d_xmin,
-1.0, "1d oscillating function with a single minimum"
],
[
\&corner4d_f, undef, 1, 4, \&corner4d_lb, \&corner4d_ub, \&corner4d_xmin,
1.0, "4d function with minimum at corner"
],
[
\&side4d_f, undef, 1, 4, \&side4d_lb, \&side4d_ub, \&side4d_xmin,
-141.285020472, "4d function with minimum at side"
],
);