# NAME

PDL::Opt::NonLinear -- Non Linear optimization routines

# SYNOPSIS

```
use PDL::Opt::NonLinear;
$x = random(5);
$gx = rosen_grad($x);
$fx = rosen($x);
$xtol = pdl(1e-16);
$gtol = pdl(0.9);
$eps = pdl(1e-10);
$print = ones(2);
$maxit = pdl(long, 200);
$info = pdl(long,0);
sub fg_func{
my ($f, $g, $x) = @_;
$f .= rosen($x);
$g .= rosen_grad($x);
}
cgfam($fx, $gx, $x, $maxit, $eps, $xtol, $gtol,$print,$info,1,\&fg_func);
```

# DESCRIPTION

This module provides routine that solves optimization problem:

```
minimize f(x)
x
```

Some routines can handle bounds, so:

```
minimize f(x)
x
subject to low <= x <= up
```

# FUNCTIONS

## tensoropt

` Signature: ([io,phys]fx();[io,phys]gx(n);[io,phys]hx(n,n);[io,phys]x(n);int [phys]method();int [io,phys]maxit();int [phys]digits();int [phys]gtype();int [phys] htype();[phys]fscale();[phys]typx(n);[phys]stepmx();[phys]xtol();[phys]gtol();int [phys]print();int [io,phys]ipr(); SV* f_func;SV* g_func;SV* h_func)`

This routine solves the optimization problem

```
minimize f(x)
x
```

where x is a vector of n real variables. The derivative tensor method method bases each iteration on a specially constructed fourth order model of the objective function. The model interpolates the function value and gradient from the previous iterate and the current function value, gradient and hessian matrix.

parameters:

```
fx --> function value and final function value
gx(n) <-- current gradient and gradient at final point
hx(n,n) --> hessian
x(n) --> initial guess (input) and final point
method --> if value is 0 then use only newton step at
each iteration, if value is 1 then try both
tensor and newton steps at each iteration
maxit <-- iteration limit and final number of iterations
digits --> number of good digits in optimization function fcn
gtype --> = 0: gradient computed by finite difference
1: analytical gradient supplied is checked
2: analytical gradient supplied
htype --> = 0: hessian computed by finite difference
1: analytical hessian supplied is checked
2: analytical hessian supplied
fscale --> estimate of scale of objective function fcn
typx(n) --> typical size of each component of x
stepmx --> maximum step length allowed
xtol --> step tolerance
gtol --> gradient tolerance
ipr --> output unit number
print --> output message control
f_func:
parameter: PDL(fx), PDL(x)
g_func:
parameter PDL(gx), PDL(x)
h_func:
parameter PDL(hx), PDL(x)
```

```
$x = random(5);
$gx = rosen_grad($x);
$hx = rosen_hess($x);
$fx = rosen($x);
$xtol = pdl(1e-16);
$gtol = pdl(1e-8);
$stepmx =pdl(0.5);
$maxit = pdl(long, 50);
sub min_func{
my ($fx, $x) = @_;
$fx .= rosen($x);
}
sub grad_func{
my ($gx, $x) = @_;
$gx .= rosen_grad($x);
}
sub hess_func{
my ($hx, $x) = @_;
$hx .= rosen_hess($x);
}
tensoropt($fx, $gx, $hx, $x,
1,$maxit,15,1,2,1,
ones(5),0.5,$xtol,$gtol,2,6,
\&min_func, \&grad_func, \&hess_func);
```

tensoropt ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## lbfgs

` Signature: ([io,phys]fx(); [io,phys]gx(n); [io,phys]x(n);[io,phys]diag(n);int [phys]diagco();int [phys]m();int [io,phys]maxit();int [io,phys]maxfc();[phys]eps();[phys]xtol();[phys]gtol();int [phys]print(2);int [io,phys]info(); SV* fg_func;SV* diag_func)`

This subroutine solves the unconstrained minimization problem

` min f(x), x= (x1,x2,...,xn),`

using the limited memory bfgs method. The routine is especially effective on problems involving a large number of variables. In a typical iteration of this method an approximation hk to the inverse of the hessian is obtained by applying m bfgs updates to a diagonal matrix hk0, using information from the previous m steps. The user specifies the number m, which determines the amount of storage required by the routine. The user may also provide the diagonal matrices hk0 if not satisfied with the default choice. The algorithm is described in "on the limited memory bfgs method for large scale optimization", by d. liu and j. nocedal, mathematical programming b 45 (1989) 503-528.

The steplength is determined at each iteration by means of the line search routine mcvsrch, which is a slight modification of the routine csrch written by Moré and Thuente.

```
where
m The number of corrections used in the bfgs update. it
is not altered by the routine. values of m less than 3 are
not recommended; large values of m will result in excessive
computing time. 3<= m <=7 is recommended. restriction: m > 0.
x On initial entry, it must be set by the user to the values
of the initial estimate of the solution vector.
On exit with info=0, it contains the values of the variables
at the best point found (usually a solution).
f is a double precision variable. before initial entry and on
a re-entry with info=1, it must be set by the user to
contain the value of the function f at the point x.
g is a double precision array of length n. before initial
entry and on a re-entry with info=1, it must be set by
the user to contain the components of the gradient g at
the point x.
diagco is a logical variable that must be set to 1 if the
user wishes to provide the diagonal matrix hk0 at each
iteration. Otherwise it should be set to 0, in which
case lbfgs will use a default value described below.
diag is a double precision array of length n. if diagco=.true.,
then on initial entry or on re-entry with info=2, diag
it must be set by the user to contain the values of the
diagonal matrix hk0. Restriction: all elements of diag
must be positive.
print is an integer array of length two which must be set by the
user.
print(1) specifies the frequency of the output:
print(1) < 0 : no output is generated,
print(1) = 0 : output only at first and last iteration,
print(1) > 0 : output every print(1) iterations.
print(2) specifies the type of output generated:
print(2) = 0 : iteration count, number of function
evaluations, function value, norm of the
gradient, and steplength,
print(2) = 1 : same as print(2)=0, plus vector of
variables and gradient vector at the
initial point,
print(2) = 2 : same as print(2)=1, plus vector of
variables,
print(2) = 3 : same as print(2)=2, plus gradient vector.
maxit On entry maximum number of iteration.
On exit, the number of iteration.
maxfc On entry maximum number of function evaluation.
On exit, the number of function evaluation.
eps is a positive double precision variable that must be set by
the user, and determines the accuracy with which the solution
is to be found. the subroutine terminates when
||g|| < eps max(1,||x||),
where ||.|| denotes the euclidean norm.
xtol is a positive double precision variable that must be set by
the user to an estimate of the machine precision (e.g.
10**(-16) on a sun station 3/60). The line search routine will
terminate if the relative width of the interval of uncertainty
is less than xtol.
gtol is a double precision variable which controls the accuracy of
the line search routine mcsrch. If the function and gradient
evaluations are inexpensive with respect to the cost of the
iteration (which is sometimes the case when solving very large
problems) it may be advantageous to set gtol to a small value.
A typical small value is 0.1. It's set to 0.9 if gtol < 1.d-04.
restriction: gtol should be greater than 1.d-04.
info is an integer variable that must be set to 0 on initial entry
to the subroutine. A return with info < 0 or info > 2 indicates
an error.
The following values of info, detecting an error,
are possible:
info=-1 the i-th diagonal element of the diagonal inverse
hessian approximation, given in diag, is not
positive.
info=-2 improper input parameters for lbfgs (n or m are
not positive).
info=-3 error in user subroutine.
if info > 2 the line search routine mcsrch failed:
info = 3 more than 20 function evaluations were
required at the present iteration.
info = 4 the step is too small.
info = 5 the step is too large.
info = 6 rounding errors prevent further progress.
there may not be a step which satisfies
the sufficient decrease and curvature
conditions. tolerances may be too small.
info = 7 relative width of the interval of
uncertainty is at most xtol.
info = 8 improper input parameters.
fg_func:
stop = fg_func PDL(fx), PDL(gx), PDL(x)
```

```
$x = random(5);
$gx = rosen_grad($x);
$fx = rosen($x);
$diag = zeroes(5);
$xtol = pdl(1e-16);
$gtol = pdl(0.9);
$eps = pdl(1e-10);
$print = ones(2);
$maxfc = pdl(long,100);
$maxit = pdl(long,50);
$info = pdl(long,0);
$diagco= pdl(long,0);
$m = pdl(long,10);
sub fdiag{};
sub fg_func{
my ($f, $g, $x) = @_;
$f .= rosen($x);
$g .= rosen_grad($x);
return 0;
}
lbfgs($fx, $gx, $x, $diag, $diagco, $m, $maxit, $maxfc, $eps, $xtol, $gtol,
$print,$info,\&fg_func,\&fdiag);
```

lbfgs ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## lbfgsb

` Signature: ([io,phys]fx(); [io,phys]gx(n); [io,phys]x(n);int [phys]m();[phys]bound(n,m=2);int [phys]tbound(n);int [io]maxit();[phys]factr();[phys]pgtol();[phys]gtol();int [phys]print(2);int [io,phys]info();int [o,phys]iv(p=44);[o,phys]v(q=29); SV* fg_func)`

This routine solves the optimization problem

```
minimize f(x)
x
subject to low <= x <= up
```

It uses the limited memory BFGS method. (The direct method will be used in the subspace minimization.)

```
x
is a double precision array of dimension n.
On entry x is an approximation to the solution.
On exit x is the current approximation.
m
On entry m is the maximum number of variable metric corrections
used to define the limited memory matrix.
On exit m is unchanged.
bound(n,2)
On entry bound(,0) is the lower bound on x.
On entry bound(,1) is the upper bound on x.
On exit bound(n,2) is unchanged.
tbound(n)
On entry nbd represents the type of bounds imposed on the
variables, and must be specified as follows:
nbd(i)=0 if x(i) is unbounded,
1 if x(i) has only a lower bound,
2 if x(i) has both lower and upper bounds, and
3 if x(i) has only an upper bound.
On exit nbd is unchanged.
fx
On first entry f is unspecified.
On final exit f is the value of the function at x.
gx(n)
On first entry g is unspecified.
On final exit g is the value of the gradient at x.
maxit
On entry maximum number of iteration.
On exit, the number of iteration
factr
On entry factr >= 0 is specified by the user. The iteration
will stop when
(f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr*epsmch
where epsmch is the machine precision, which is automatically
generated by the code. Typical values for factr: 1.d+12 for
low accuracy; 1.d+7 for moderate accuracy; 1.d+1 for extremely
high accuracy.
pgtol
On entry pgtol >= 0 is specified by the user. The iteration
will stop when
max{|proj g_i | i = 1, ..., n} <= pgtol
where pg_i is the ith component of the projected gradient.
gtol
Controls the accuracy of the line search routine mcsrch.
If the function and gradient evaluations are inexpensive with
respect to the cost of the iteration (which is sometimes the case
when solving very large problems) it may be advantageous to set
gtol to a small value. A typical small value is 0.1.
It's set to 0.9 if gtol < 1.d-04.
Restriction: gtol should be greater than 1.d-04.
print
Controls the frequency and type of output generated:
print[0] < 0 no output is generated;
print[0] = 0 print only one line at the last iteration;
0 < print[0] < 99 print also f and |proj g| every iprint iterations;
print[0] = 99 print details of every iteration except n-vectors;
print[0] = 100 print also the changes of active set and final x;
print[0] > 100 print details of every iteration including x and g;
When print[1] > 0, the file iterate.dat will be created to
summarize the iteration.
info
On entry 0,
On exit, contain error code:
0 : no error
-1: the routine has terminated abnormally
without being able to satisfy the termination conditions,
x contains the best approximation found,
f and g contain f(x) and g(x) respectively
-2: the routine has detected an error in the
input parameters;
iv(44)
On exit, at end of an iteration, the following information is
available:
iv(21) = the total number of intervals explored in the
search of Cauchy points;
iv(25) = the total number of skipped BFGS updates before
the current iteration;
iv(29) = the number of current iteration;
iv(30) = the total number of BFGS updates prior the current
iteration;
iv(32) = the number of intervals explored in the search of
Cauchy point in the current iteration;
iv(33) = the total number of function and gradient
evaluations;
iv(35) = the number of function value or gradient
evaluations in the current iteration;
if iv(36) = 0 then the subspace argmin is within the box;
if iv(36) = 1 then the subspace argmin is beyond the box;
iv(37) = the number of free variables in the current
iteration;
iv(38) = the number of active constraints in the current
iteration;
n + 1 - iv(39) = the number of variables leaving the set of
active constraints in the current iteration;
iv(40) = the number of variables entering the set of active
constraints in the current iteration.
else
iv(29) = the current iteration number;
iv(33) = the total number of function and gradient
evaluations;
iv(35) = the number of function value or gradient
evaluations in the current iteration;
iv(37) = the number of free variables in the current
iteration;
iv(38) = the number of active constraints at the current
iteration
v(29)
On exit, at end of an iteration, the following information is
available:
v(0) = current 'theta' in the BFGS matrix;
v(1) = f(x) in the previous iteration;
v(2) = factr*epsmch;
v(3) = 2-norm of the line search direction vector;
v(4) = the machine precision epsmch generated by the code;
v(6) = the accumulated time spent on searching for Cauchy points;
v(7) = the accumulated time spent on subspace minimization;
v(8) = the accumulated time spent on line search;
v(10) = the slope of the line search function at
the current point of line search;
v(11) = the maximum relative step length imposed in line search;
v(12) = the infinity norm of the projected gradient;
v(13) = the relative step length in the line search;
v(14) = the slope of the line search function at the starting point of the line search;
v(15) = the square of the 2-norm of the line search direction vector.
scalar fg_func: computes the value(fx) and gradient(gx) of the function at x.
iv and v are also provided for info
param fx, gx, x, iv, v
return value
-1 stop now and restore the information at
the latest iterate
0 continue
1 last iteration
```

```
# Global Optimization
# Try to solve (with threading)
# The SIAM 100-Digit Challenge problem 4
# see http://www-m8.ma.tum.de/m3/bornemann/challengebook/
# result: -3.30686864747523728007611377089851565716648236
use PDL::Opt::NonLinear;
use PDL::Stat::Distributions;
$x = (random(2,500)-0.5)*2;
$gx = zeroes(2,500);
$fx = zeroes(500);
$bounds = zeroes(2,2);
$bounds(,0).= -1;
$bounds(,1).= 1;
$tbounds = zeroes(2);
$tbounds .= 2;
$gtol = pdl(0.9);
$pgtol = pdl(1e-4);
$factr = pdl(10000);
$m = pdl(10);
$print = pdl([-1,0]);
$maxit = zeroes(long,500);
$maxit .= 200;
$info = zeroes(long,500);
$iv = zeroes(long,44,500);
$v = zeroes(29,500);
sub fg_func{
my ($f, $g, $x) = @_;
$f.= exp(sin(50*$x(0)))+sin(60*exp($x(1)))+
sin(70*sin($x(0)))+sin(sin(80*$x(1)))-
sin(10*($x(0)+$x(1)))+($x(0)**2+$x(1)**2)/4;
$g(0) .= 50*cos(50*$x(0))* exp(sin(50*$x(0)))+
70*cos(70*sin($x(0)))*cos($x(0))-
10*cos(10*$x(0)+10*$x(1))+1/2*$x(0);
$g(1) .= 60*cos(60*exp($x(1)))* exp($x(1))+
80*cos(sin(80*$x(1)))* cos(80*$x(1))-
10*cos(10*$x(0)+10*$x(1))+1/2*$x(1);
return 0;
}
lbfgsb($fx, $gx, $x, $m, $bounds, $tbounds, $maxit, $factr, $pgtol, $gtol,
$print, $info,$iv, $v,\&fg_func);
print $fx->min;
# Local Optimization
$x = random(5);
$gx = zeroes(5);
$fx = pdl(0);
$bounds = zeroes(5,2);
$bounds(,0).= -5;
$bounds(,1).= 5;
$tbounds = zeroes(5);
$tbounds .= 2;
$gtol = pdl(0.9);
$pgtol = pdl(1e-10);
$factr = pdl(100);
$print = pdl(long, [1,0]);
$maxit = pdl(long,100);
$info = pdl(long,0);
$m = pdl(long,10);
$iv = zeroes(long,44);
$v = zeroes(29);
sub fg_func{
my ($f, $g, $x) = @_;
$f .= rosen($x);
$g .= rosen_grad($x);
return 0;
}
lbfgsb($fx, $gx, $x, $m, $bounds, $tbounds, $maxit, $factr, $pgtol, $gtol,
$print, $info,$iv, $v,\&fg_func);
```

lbfgsb ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## spg

` Signature: ([io,phys]fx();[io,phys]x(n);int [io,phys]m();int [io,phys]maxit();int [phys]maxfc();[phys]eps1();[phys]eps2();int [phys]print();int [io,phys]fcnt();int [io,phys]gcnt();[io,phys]pginf();[io,phys]pgtwon();int [io,phys]info(); SV* min_func; SV* grad_func; SV* px_func)`

This routine solves the optimization problem

```
minimize f(x)
x
```

where x is a vector of n real variables. The method used is a Spectral Projected Gradient (Version 2: "continuous projected gradient direction") to find the local minimizers of a given function with convex constraints, described in E. G. Birgin, J. M. Martinez, and M. Raydan, "Nonmonotone spectral projected gradient methods on convex sets", SIAM Journal on Optimization 10, pp. 1196-1211, 2000. and E. G. Birgin, J. M. Martinez, and M. Raydan, "SPG: software for convex-constrained optimization", ACM Transactions on Mathematical Software, 2001 (to appear).

The user must supply the external subroutines evalf, evalg and proj to evaluate the objective function and its gradient and to project an arbitrary point onto the feasible region.

This version 17 JAN 2000 by E.G.Birgin, J.M.Martinez and M.Raydan. Reformatted 03 OCT 2000 by Tim Hopkins. Final revision 03 JUL 2001 by E.G.Birgin, J.M.Martinez and M.Raydan.

```
On Entry:
x(n) initial guess,
m number of previous function values to be considered
in the nonmonotone line search,
eps1 stopping criterion: ||projected grad||_inf < eps,
eps2 stopping criterion: ||projected grad||_2 < eps2,
maxit integer,
maximum number of iterations,
maxfc integer,
maximum number of function evaluations,
print logical,
true: print some information at each iteration,
false: no print.
On Return:
x(n) approximation to the local minimizer,
fx: function value at the approximation to the local
minimizer,
pginfn
||projected grad||_inf at the final iteration,
pgtwon
||projected grad||_2^2 at the final iteration,
maxit
number of iterations,
fcnt number of function evaluations,
gcnt number of gradient evaluations,
info termination parameter:
0= convergence with projected gradient infinite-norm,
1= convergence with projected gradient 2-norm,
2= too many iterations,
3= too many function evaluations,
4= error in proj subroutine,
5= error in evalf subroutine,
6= error in evalg subroutine.
min_func:
parameter: PDL(fx), PDL(x)
grad_func:
parameter: PDL(gx), PDL(x)
px_func:
parameter: PDL(x)
```

```
# Bounded example
$bounds = zeroes(5,2);
$bounds(,0) .= -5;
$bounds(,1) .= 5;
$info = pdl(long,0);
$print = pdl(long,1);
$fcnt = pdl(long,0);
$gcnt = pdl(long,0);
$pginf = pdl(0);
$pgtwon = pdl(0);
$maxit = pdl(long , 500);
$maxfc = pdl(long , 1000);
$m = pdl(long , 100);
$eps1 = pdl(0);
$eps2 = pdl(1e-5);
$fx = pdl(0);
$a= random(5)
sub pgrad{
$x = shift;
$c = minimum transpose(cat $x, $bounds(,1));
$c = maximum transpose(cat $c, $bounds(,0));
$x .=$c;
return 0;
}
sub grad{
($aa, $bb) = @_;
$aa .= rosen_grad($bb);
return 0;
}
sub min_func{
($aa, $bb) = @_;
$aa .= rosen($bb);
return 0;
}
spg($fx, $a, $m, $maxit, $maxfc, $eps1, $eps2, $print, $fcnt, $gcnt, $pginf, $pgtwon, $info, \&min_func,\&grad, \&pgrad);
```

spg ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## lmqn

` Signature: ([io,phys]fx();[io,phys]gx(n);[io,phys]x(n);int [io,phys]maxit();int [io,phys]maxfc();int [phys]cgmaxit();[phys]xtol();[phys]accrcy();[phys]eta();[phys]stepmx();int [phys]print();int [io,phys]info(); SV* fg_func)`

This routine solves the optimization problem

```
minimize f(x)
x
```

where x is a vector of n real variables. The method used is a truncated-newton algorithm (see "newton-type minimization via the lanczos method" by s.g. nash (siam j. numer. anal. 21 (1984), pp. 770-778). This algorithm finds a local minimum of f(x). It does not assume that the function f is convex (and so cannot guarantee a global solution), but does assume that the function is bounded below. It can solve problems having any number of variables, but it is especially useful when the number of variables (n) is large.

```
subroutine parameters:
fx On input, a rough estimate of the value of the
objective function at the solution; on output, the value
of the objective function at the solution
gx(n) on output, the final value of the gradient
x(n) on input, an initial estimate of the solution;
on output, the computed solution.
maxit maximum number of inner iterations
maxfc maximum allowable number of function evaluations
maxit maximum number of inner iterations per step
cgmaxit maximum number of inner iterations per step
(preconditionned conjugate iteration)
eta severity of the linesearch
xtol desired accuracy for the solution x*
stepmx maximum allowable step in the linesearch
accrcy accuracy of computed function values
print determines quantity of printed output
0 = none, 1 = one line per major iteration.
info ( 0 => normal return)
( 1 => more than maxit iterations)
( 2 => more than maxfun evaluations)
( 3 => line search failed to find
( lower point (may not be serious)
(-1 => error in input parameters)
fg_func:
parameter: PDL(fx), PDL(gx), PDL(x)
```

```
$x = random(5);
$gx = $x->zeroes;
$fx = rosen($x);
$accrcy = pdl(1e-16);
$xtol = pdl(1e-10);
$stepmx =pdl(1);
$eta =pdl(0.9);
$info = pdl(long, 0);
$print = pdl(long, 1);
$maxit = pdl(long, 50);
$cgmaxit = pdl(long, 50);
$maxfc = pdl(long,250);
sub fg_func{
my ($f, $g, $x) = @_;
$f .= rosen($x);
$g .= rosen_grad($x);
}
lmqn($fx, $gx, $x, $maxit, $maxfc, $cgmaxit, $xtol, $accrcy, $eta, $stepmx, $print, $info,\&fg_func);
```

lmqn ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## lmqnbc

` Signature: ([io,phys]fx();[io,phys]gx(n);[io,phys]x(n);[phys]bound(n,m=2);int [io,phys]maxit();int [io,phys]maxfc();int [phys]cgmaxit();[phys]xtol();[phys]accrcy();[phys]eta();[phys]stepmx();int [phys]print();int [io,phys]info(); SV* fg_func)`

This routine solves the optimization problem

```
minimize f(x)
x
subject to low <= x <= up
```

where x is a vector of n real variables. The method used is a truncated-newton algorithm (see "newton-type minimization via the lanczos algorithm" by s.g. nash (technical report 378, math. The lanczos method" by s.g. nash (siam j. numer. anal. 21 (1984), pp. 770-778). This algorithm finds a local minimum of f(x). It does not assume that the function f is convex (and so cannot guarantee a global solution), but does assume that the function is bounded below. It can solve problems having any number of variables, but it is especially useful when the number of variables (n) is large.

```
subroutine parameters:
fx On input, a rough estimate of the value of the
objective function at the solution; on output, the value
of the objective function at the solution
gx(n) on output, the final value of the gradient
x(n) on input, an initial estimate of the solution;
on output, the computed solution.
bound(n,2)
The lower and upper bounds on the variables. if
there are no bounds on a particular variable, set
the bounds to -1.d38 and 1.d38, respectively.
maxit maximum number of inner iterations
maxfc maximum allowable number of function evaluations
cgmaxit maximum number of inner iterations per step
(preconditionned conjugate iteration)
eta severity of the linesearch
xtol desired accuracy for the solution x*
stepmx maximum allowable step in the linesearch
accrcy accuracy of computed function values
print determines quantity of printed output
0 = none, 1 = one line per major iteration.
info ( 0 => normal return)
( 1 => more than maxit iterations)
( 2 => more than maxfun evaluations)
( 3 => line search failed to find
( lower point (may not be serious)
(-1 => error in input parameters)
fg_func:
parameter: PDL(fx), PDL(gx), PDL(x)
```

```
$x = random(5);
$gx = $x->zeroes;
$fx = rosen($x);
$bounds = zeroes(5,2);
$bounds(,0).= -5;
$bounds(,1).= 5;
$accrcy = pdl(1e-20);
$xtol = pdl(1e-10);
$stepmx =pdl(1);
$eta = pdl(0.9);
$info = pdl(long, 0);
$print = pdl(long, 1);
$maxit = pdl(long, 100);
$maxfc = pdl(long,250);
$cgmaxit = pdl(long, 50);
sub fg_func{
my ($f, $g, $x) = @_;
$f .= rosen($x);
$g .= rosen_grad($x);
}
lmqnbc($fx, $gx, $x, $bounds, $maxit, $maxfc, $cgmaxit, $xtol, $accrcy, $eta, $stepmx, $print, $info,\&fg_func);
```

lmqnbc ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## cgfam

` Signature: ([io,phys]fx(); [io,phys]gx(n); [io,phys]x(n);int [io,phys]maxit();[phys]eps();[io,phys]xtol();[io,phys]gtol();int [phys]print(2);int [io,phys]info(); int [phys]method(); SV* fg_func)`

This subroutine solves the unconstrained minimization problem

` min f(x), x= (x1,x2,...,xn),`

using conjugate gradient methods, as described in the paper: gilbert, j.c. and nocedal, j. (1992). "global convergence properties of conjugate gradient methods", siam journal on optimization, vol. 2, pp. 21-42.

```
where
fx is a double precision variable. before initial entry and on
a re-entry with info=1, it must be set by the user to
contain the value of the function f at the point x.
gx is a double precision array of length n. before initial
entry and on a re-entry with info=1, it must be set by
the user to contain the components of the gradient g at
the point x.
x on initial entry, it must be set by the user to the values
of the initial estimate of the solution vector.
on exit with info=0, it contains the values of the variables
at the best point found (usually a solution).
maxit maximum number of iterations.
eps is a positive double precision variable that must be set by
the user, and determines the accuracy with which the solution
is to be found. the subroutine terminates when
||g|| < eps max(1,||x||),
where ||.|| denotes the euclidean norm.
xtol is a positive double precision variable that must be set by
the user to an estimate of the machine precision (e.g.
10**(-16) on a sun station 3/60). the line search routine will
terminate if the relative width of the interval of uncertainty
is less than xtol.
gtol is a double precision variable which controls the accuracy of
the line search routine mcsrch. if the function and gradient
evaluations are inexpensive with respect to the cost of the
iteration (which is sometimes the case when solving very large
problems) it may be advantageous to set gtol to a small value.
A typical small value is 0.1. It's set to 0.9 if gtol < 1.d-04.
restriction: gtol should be greater than 1.d-04.
print frequency and type of printing
iprint(1) < 0 : no output is generated
iprint(1) = 0 : output only at first and last iteration
iprint(1) > 0 : output every iprint(1) iterations
iprint(2) : specifies the type of output generated;
the larger the value (between 0 and 3),
the more information
iprint(2) = 0 : no additional information printed
iprint(2) = 1 : initial x and gradient vectors printed
iprint(2) = 2 : x vector printed every iteration
iprint(2) = 3 : x vector and gradient vector printed
every iteration
info controls termination of code, and return to main
program to evaluate function and gradient
info = -3 : improper input parameters
info = -2 : descent was not obtained
info = -1 : line search failure
info = 0 : initial entry or
successful termination without error
info = 1 : user canceled optimization (maximum iteration)
info = 2 : user canceled optimization
method = 1 : fletcher-reeves
2 : polak-ribiere
3 : positive polak-ribiere ( beta=max{beta,0} )
scalar fg_func: computes the value(fx) and gradient(gx) of the function at x.
param fx, gx, x
return value
0 continue
1 last iteration
```

```
$x = random(5);
$gx = rosen_grad($x);
$fx = rosen($x);
$xtol = pdl(1e-10);
$gtol = pdl(0.9);
$eps = pdl(1e-10);
$print = ones(2);
$maxit = pdl(long, 200);
$info = pdl(long,0);
sub fg_func{
my ($f, $g, $x) = @_;
$f .= rosen($x);
$g .= rosen_grad($x);
return 0;
}
cgfam($fx, $gx, $x, $maxit, $eps, $xtol, $gtol,$print,$info,1,\&fg_func);
```

cgfam ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## hooke

` Signature: ([io,phys]x(n);int [io,phys]maxit();[phys]rho();[phys]tol(); SV* hooke_func)`

Find a point X where the nonlinear function f(X) has a local minimum. X is an n-vector and f(X) is a scalar. In mathematical notation

` f: R^n -> R^1. `

The objective function f() is not required to be continuous. Nor does f() need to be differentiable. The program does not use or require derivatives of f().

The software user supplies three things: a subroutine that computes f(X), an initial "starting guess" of the minimum point X, and values for the algorithm convergence parameters. Then the program searches for a local minimum, beginning from the starting guess, using the Direct Search algorithm of Hooke and Jeeves.

rho controls convergence :

The algorithm works by taking "steps" from one estimate of a minimum, to another (hopefully better) estimate. Taking big steps gets to the minimum more quickly, at the risk of "stepping right over" an excellent point. The stepsize is controlled by a user supplied parameter called rho. At each iteration, the stepsize is multiplied by rho (0 < rho < 1), so the stepsize is successively reduced. Small values of rho correspond to big stepsize changes, which make the algorithm run more quickly. However, there is a chance (especially with highly nonlinear functions) that these big changes will accidentally overlook a promising search vector, leading to nonconvergence. Large values of rho correspond to small stepsize changes, which force the algorithm to carefully examine nearby points instead of optimistically forging ahead. This improves the probability of convergence. The stepsize is reduced until it is equal to (or smaller than) tol. So the number of iterations performed by Hooke-Jeeves is determined by rho and tol:

` rho**(number_of_iterations) = tol `

In general it is a good idea to set rho to an aggressively small value like 0.5 (hoping for fast convergence). Then, if the user suspects that the reported minimum is incorrect (or perhaps not accurate enough), the program can be run again with a larger value of rho such as 0.85, using the result of the first minimization as the starting guess to begin the second minimization.

```
x: On entry this is the user-supplied guess at the minimum.
On exit this is the location of the local minimum,
calculated by the program
maxit On entry, a rarely used, halting
criterion. If the algorithm uses >= maxit
iterations, halt.
On exit number of iteration.
rho This is a user-supplied convergence
parameter (more detail above), which should be
set to a value between 0.0 and 1.0. Larger
values of rho give greater probability of
convergence on highly nonlinear functions, at a
cost of more function evaluations. Smaller
values of rho reduces the number of evaluations
(and the program running time), but increases
the risk of nonconvergence. See below.
tol This is the criterion for halting
the search for a minimum. When the algorithm
begins to make less and less progress on each
iteration, it checks the halting criterion: if
the stepsize is below tol, terminate the
iteration and return the current best estimate
of the minimum. Larger values of tol (such
as 1.0e-4) give quicker running time, but a
less accurate estimate of the minimum. Smaller
values of tol (such as 1.0e-7) give longer
running time, but a more accurate estimate of
the minimum.
func objective function to be minimized.
scalar double fun ($x(n))
```

```
$x = random(2);
sub test{
my $a = shift;
rosen($a)->sclr;
}
$rho = pdl(0.5);
$tol = pdl(1e-7);
$maxit =pdl(long, 500);
$x->hooke($maxit, $rho,$tol,\&test);
print "Minimum found at $x in $maxit iteration(s)";
```

hooke ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## gencan

```
Signature: ([io,phys]fx();[io,phys]gx(n);[io,phys]x(n);[phys]bound(n,m=2);[phys]fmin();int [phys]maxit();int [phys]maxfc();
int [phys]nearlyq();int [phys]gtype();int [phys]htvtype();int [phys]trtype();int [phys]fmaxit();int [phys]gmaxit();
int [phys]interpmaxit();int [phys]cgstop();int [phys]cgmaxit();int [phys]qmpmaxit();
[phys]ftol();[phys]epsgpen();[phys]epsgpsn();[phys]cggtol();[phys]cgitol();[phys]cgftol();
[phys]qmptol();[phys]delta();[phys]eta();[phys]delmin();
[phys]lammin();[phys]lammax();[phys]theta();[phys]gamma();[phys]beta();
[phys]sigma1();[phys]sigma2();[phys]nint();[phys]next();
[phys]sterel();[phys]steabs();[phys]epsrel();[phys]epsabs();[phys]infty();
[o,phys]gpeucn2();[o,phys]gpsupn();int [o,phys]iter();int [o,phys]fcnt();int [o,phys]gcnt();int [o,phys]cgcnt();
int [o,phys]spgiter();int [o,phys] spgfcnt();int [o,phys]tniter();int [o,phys]tnfcnt();int [o,phys]tnstpcnt();
int [o,phys]tnintcnt();int [o,phys] tnexgcnt();int [o,phys]tnexbcnt();int [o,phys]tnintfe();int [o,phys]tnexgfe();int [o,phys]tnexbfe();
int [phys]print(p);int [phys]ncomp();int [io,phys]info(); SV* f_func; SV* g_func; SV* h_func)
```

Solves the box-constrained minimization problem

```
Minimize f(x)
subject to l \leq x \leq u
```

using a method described in E. G. Birgin and J. M. Martinez, "Large-scale active-set box-constrained optimization method with spectral projected gradients", Computational Optimization and Applications 23, 101-125 (2002).

Subroutines evalf and evalg must be supplied by the user to evaluate the function f and its gradient, respectively. The calling sequences are

```
inform evalf(f, x)
inform evalg(g, x)
```

where x is the point where the function (the gradient) must be evaluated, n is the number of variables and f (g) is the functional value (the gradient vector). The real parameters x, f, g must be double precision.

A subroutine evalhd to compute the Hessian times vector products is optional. If this subroutine is not provided an incremental quotients version will be used instead. The calling sequence of this subroutine should be

` inform call evalhd(hu, x, u, ind)`

where x is the point where the approx-Hessian is being considered, u is the vector which should be multiplied by the approx-Hessian H and hu is the vector where the product should be placed. The information about the matrix H must be passed to evalhd by means of common declarations. The necessary computations must be done in evalg. The real parameters x, u, hu must be double precision.

This subroutine must be coded by the user, taking into account that n is the number of variables of the problem and that hu must be the product H u. Moreover, you must assume, when you code evalhd, that only size(ind) components of u are nonnull and that ind is the set of indices of those components. In other words, you must write evalhd in such a way that hu is the vector whose i-th entry is

` hu(i) = \Sum_{j=1}^{nind} H_{i,ind(j)} u_ind(j)`

Moreover, the only components of hu that you need to compute are those which corresponds to the indices ind(1),...,ind(nind). However, observe that you must assume that, in u, the whole vector is present, with its n components, even the zeroes. So, if you decide to code evalhd without taking into account the presence of ind and nind, you can do it. A final observation: probably, if nind is close to n, it is not worthwhile to use ind, due to the cost of accessing the correct indices. If you want, you can test, within your evalhd, if (say) nind > n/2, and, in this case use a straightforward scalar product for the components of hu.

Example: Suppose that the matrix H is full. The main steps of evalhd could be:

```
do i= 1, nind
indi= ind(i)
hu(indi)= 0.0d0
do j= 1, nind
indj= ind(j)
hu(indi)= hu(indi) + H(indi,indj) * u(indj)
end do
end do
On Entry
x double precision x(n)
initial estimate to the solution
bounds(n,2)
lower bounds and upper bounds
epsgpen double precision
small positive number for declaring convergence when the
euclidian norm of the projected gradient is less than
or equal to epsgpen
RECOMMENDED: epsgpen = 1.0d-5
epsgpsn double precision
small positive number for declaring convergence when the
infinite norm of the projected gradient is less than
or equal to epsgpsn
RECOMMENDED: epsgpsn = 1.0d-5
ftol double precision
'lack of enough progress' measure. The algorithm stops by
'lack of enough progress' when f(x_k) - f(x_{k+1}) <=
ftol * max { f(x_j)-f(x_{j+1}, j<k} during fmaxit
consecutive iterations. This stopping criterion may be
inhibited setting ftol = 0. We recommend, preliminary, to
set ftol = 0.01 and fmaxit = 5
RECOMMENDED: ftol = 1.0d-2
fmaxit integer
see the meaning of ftol, above
RECOMMENDED: fmaxit = 5
gmaxit integer
If the order of the euclidian-norm of the continuous projected
gradient did not change during gmaxit consecutive iterations
the execution stops. Recommended: gmaxit= 10. In any case
gmaxit must be greater than or equal to 1
RECOMMENDED: gmaxit = 10
fmin double precision
function value for the stopping criteria f <= fmin
RECOMMENDED: fmin = -1.0d+99 (inhibited)
maxit integer
maximum number of iterations allowed
RECOMMENDED: maxit = 1000
maxfc integer
maximum number of funtion evaluations allowed
RECOMMENDED: maxfc = 5000
delta
initial trust-region radius. Default max{0.1||x||,0.1} is set
if you set delta < 0. Otherwise, the parameters delta
will be the ones set by the user.
RECOMMENDED: delta = -1
cgmaxit integer
maximum number of iterations allowed for the cg subalgorithm
Default values for this parameter and the previous one are
0.1 and 10 * log (number of free variables). Default values
are taken if you set ucgeps < 0 and cgmaxit < 0,
respectively. Otherwise, the parameters ucgeps and cgmaxit
will be the ones set by the user
RECOMMENDED: cgmaxit = -1
cgstop, cggtol double precision
cgstop means cunjugate gradient stopping criterion relation, and
cggtol means conjugate gradients projected gradient final norm.
Both are related to a stopping criterion of conjugate gradients.
This stopping criterion depends on the norm of the residual
of the linear system. The norm of the this residual should be
less or equal than a 'small'quantity which decreases as we are
approximating the solution of the minimization problem (near the
solution, better the truncated-Newton direction we aim). Then, the
log of the required precision requested to conjugate gradient has
a linear dependence on the log of the norm of the projected
gradient. This linear relation uses the squared euclidian-norm
of the projected gradient if cgstop = 1 and uses the sup-norm if
cgstop = 2. In adition, the precision required to CG is equal to
cgitol (conjugate gradient initial epsilon) at x0 and cgftol
(conjugate gradient final epsilon) when the euclidian- or sup-norm
of the projected gradient is equal to cggtol (conjugate gradients
projected gradient final norm) which is an estimation of the value
of the euclidian- or sup-norm of the projected gradient at the
solution.
RECOMMENDED: cgstop = 1, cggtol = epsgpen; or
cgstop = 2, cggtol = epsgpsn.
cgitol, cgftol double precision
small positive numbers for declaring convergence of the
conjugate gradient subalgorithm when
||r||_2 < cgeps * ||rhs||_2, where r is the residual and
rhs is the right hand side of the linear system, i.e., cg
stops when the relative error of the solution is smaller
that cgeps.
cgeps varies from cgitol to cgftol in such a way that, depending
on cgstop (see above),
i) log10(cgeps^2) depends linearly on log10(||g_P(x)||_2^2)
which varies from ||g_P(x_0)||_2^2 to epsgpen^2; or
ii) log10(cgeps) depends linearly on log10(||g_P(x)||_inf)
which varies from ||g_P(x_0)||_inf to epsgpsn.
RECOMMENDED: cgitol = 1.0d-1, cgftol = 1.0d-5
qmptol double precision
see below
qmpmaxit integer
This and the previous one parameter are used for a stopping
criterion of the conjugate gradient subalgorithm. If the
progress in the quadratic model is less or equal than a
fraction of the best progress ( qmptol * bestprog ) during
qmpmaxit consecutive iterations then CG is stopped by not
enough progress of the quadratic model.
RECOMMENDED: qmptol = 1.0d-4, qmpmaxit = 5
nearlyq logical
if function f is (nearly) quadratic, use the option
nearlyq = 0 Otherwise, keep the default option.
if, at an iteration of CG we find a direction d such
that d^T H d <= 0 then we take the following decision:
(i) if nearlyq = 1 then take direction d and try to
go to the boundary chosing the best point among the two
point at the boundary and the current point.
(ii) if nearlyq = 0 then we stop at the current point.
RECOMMENDED: nearlyq = 0
gtype integer
type of gradient calculation
gtype = 0 means user suplied evalg subroutine,
gtype = 1 means central diference approximation.
RECOMMENDED: gtype = 0
(provided you have the evalg subroutine)
htvtype integer
type of gradient calculation
htvtype = 0 means user suplied evalhd subroutine,
htvtype = 1 means incremental quotients approximation.
RECOMMENDED: htvtype = 1
(you take some risk using this option but, unless you have
a good evalhd subroutine, incremental quotients is a very
cheap option)
trtype integer
type of trust-region radius
trtype = 0 means 2-norm trust-region
trtype = 1 means infinite-norm trust-region
RECOMMENDED: trtype = 0
print(0) integer
commands printing. Nothing is printed if print < 0.
If print = 0, only initial and final information is printed.
If print > 0, information is printed every print iterations.
Exhaustive printing when print > 0 is commanded by print(1).
RECOMMENDED: print(0) = 1
print(1) integer
When print(0) > 0, detailed printing can be required setting
print(1) = 1.
RECOMMENDED: print(1) = 1
eta double precision
constant for deciding abandon the current face or not
We abandon the current face if the norm of the internal
gradient (here, internal components of the continuous
projected gradient) is smaller than (1-eta) times the
norm of the continuous projected gradient. Using eta=0.9
is a rather conservative strategy in the sense that
internal iterations are preferred over SPG iterations.
RECOMMENDED: eta = 0.9
delmin double precision
minimum 'trust region' to compute the Truncated Newton
direction
RECOMMENDED: delmin = 0.1
lammin, lammax double precision
The spectral steplength, called lambda, is projected
inside the box [lammin,lammax]
RECOMMENDED: lammin = 10^{-10} and lammax = 10^{10}
theta double precision
constant for the angle condition, i.e., at iteration k
we need a direction d_k such that
<g_k,d_k> <= -theta ||g||_2 ||d_k||_2,
where g_k is \nabla f(x_k)
RECOMMENDED: theta = 10^{-6}
gamma double precision
constant for the Armijo crtierion
f(x + alpha d) <= f(x) + gamma * alpha * <\nabla f(x),d>
RECOMMENDED: gamma = 10^{-4}
beta double precision
constant for the beta condition
<d_k, g(x_k + d_k)> .ge. beta * <d_k,g_k>
if (x_k + d_k) satisfies the Armijo condition but does not
satisfy the beta condition then the point is accepted, but
if it satisfied the Armijo condition and also satisfies the
beta condition then we know that there is the possibility
for a succesful extrapolation
RECOMMENDED: beta = 0.5
sigma1, sigma2 double precision
constant for the safeguarded interpolation
if alpha_new \notin [sigma1, sigma*alpha] then we take
alpha_new = alpha / nint
RECOMMENDED: sigma1 = 0.1 and sigma2 = 0.9
nint double precision
constant for the interpolation. See the description of
sigma1 and sigma2 above. Sometimes we take as a new trial
step the previous one divided by nint
RECOMMENDED: nint = 2.0
next double precision
constant for the extrapolation
when extrapolating we try alpha_new = alpha * next
RECOMMENDED: next = 2.0
interpmaxit integer
constant for testing if, after having made at least interpmaxit
interpolations, the steplength is too small. In that case
failure of the line search is declared (may be the direction
is not a descent direction due to an error in the gradient
calculations)
RECOMMENDED: interpmaxit = 4
(use interpmaxit > maxfc for inhibit this stopping criterion)
ncomp integer
this constant is just for printing. In a detailed printing
option, ncomp component of the actual point will be printed
RECOMMENDED: ncomp = 5
sterel, steabs double precision
this constants mean a 'relative small number' and 'an
absolute small number' for the increments in finite
difference approximations of derivatives
RECOMMENDED: epsrel = 10^{-7}, epsabs = 10^{-10}
epsrel, epsabs, infty double precision
this constants mean a 'relative small number', 'an
absolute small number', and 'infinite or a very big
number'. Basically, a quantity A is considered negligeble
with respect to another quantity B if
|A| < max ( epsrel * |B|, epsabs )
RECOMMENDED: epsrel = 10^{-10}, epsabs = 10^{-20} and
infty = 10^{+20}
On Return
x double precision x(n)
final estimation to the solution
f double precision
function value at the final estimation
g double precision g(n)
gradient at the final estimation
gpeucn2 double precision
squared 2-norm of the continuous projected
gradient g_p at the final estimation (||g_p||_2^2)
gpsupn double precision
||g_p||_inf at the final estimation
iter integer
number of iterations
fcnt integer
number of function evaluations
gcnt integer
number of gradient evaluations
cgcnt integer
number of conjugate gradient iterations
spgiter integer
number of SPG iterations
spgfcnt integer
number of function evaluations in SPG-directions line searches
tniter integer
number of Truncated Newton iterations
tnfcnt integer
number of function evaluations in TN-directions line searches
tnintcnt integer
number of times a backtracking in a TN-direction was needed
tnexgcnt integer
number of times an extrapolation in a TN-direction was
successfull in decreass the function value
tnexbcnt integer
number of times an extrapolation was aborted in the first
extrapolated point by augment of the function value
info
This output parameter tells what happened in this
subroutine, according to the following conventions:
0= convergence with small euclidian-norm of the
projected gradient (smaller than epsgpen);
1= convergence with small infinite-norm of the
projected gradient (smaller than epsgpsn);
2= the algorithm stopped by 'lack of enough progress',
that means that f(x_k) - f(x_{k+1}) <=
ftol * max { f(x_j)-f(x_{j+1}, j<k} during fmaxit
consecutive iterations;
3= the algorithm stopped because the order of the euclidian-
norm of the continuous projected gradient did not change
during gmaxit consecutive iterations. Probably, we
are asking for an exagerately small norm of continuous
projected gradient for declaring convergence;
4= the algorithm stopped because the functional value
is very small (f <= fmin);
6= too small step in a line search. After having made at
least interpmaxit interpolations, the steplength becames
small. 'small steplength' means that we are at point
x with direction d and step alpha, and
alpha * ||d||_infty < max(epsabs, epsrel * ||x||_infty).
In that case failure of the line search is declared
(may be the direction is not a descent direction
due to an error in the gradient calculations). Use
interpmaxit > maxfc for inhibit this criterion;
7= it was achieved the maximum allowed number of
iterations (maxit);
8= it was achieved the maximum allowed number of
function evaluations (maxfc);
9= error in evalf subroutine;
10= error in evalg subroutine;
11= error in evalhd subroutine.
```

```
$x = random(50);
$gx = $x->zeroes;
$fx = pdl(0);
$print = pdl(long,[1,0]);
$info = pdl(long,0);
$bounds = zeroes(50,2);
$bounds(,0).=-5;
$bounds(,1).=5;
sub f_func{
my ($fx, $x) = @_;
$fx .= rosen($x);
return 0;
}
sub g_func{
my ($gx, $x) = @_;
$gx .= rosen_grad($x);
return 0;
}
sub h_func{
my ($hx, $x, $d, $ind) = @_;
$hx .= rosen_hess($x,1) x $d;
return 0;
}
gencan($fx, $gx, $x, $bounds, -1e308, 200, 1000,
1, 0, 0, 0, 5, 10, 5, 1, -1, 5,
0, 1e-10, 1e-10, 1e-8, 0.1, 1e-8, 1e-8,
-1, 0.9, 0.1,
1e+40, 1e-40, 1e-6, 0.0001, 0.5, 0.1,0.9,
2, 2, 1e-10, 1e-99, 1e-30, 1e-99, 1e+308,
($gpeucn2=null), ($gpsupn=null), ($iter=null), ($fcnt=null),
($gcnt=null), ($cgcnt=null), ($spgiter=null), ($spgfcnt=null),
($tniter=null), ($tnfcnt=null), ($tnstpcnt=null), ($tnintcnt=null), ($tnexgcnt=null), ($tnexbcnt=null),
($tnintfe=null), ($tnexgfe=null), ($tnexbfe=null),
$print,5, $info,\&f_func,\&g_func, \&h_func);
```

gencan ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## sgencan

```
Signature: ([io,phys]fx();[io,phys]gx(n);[io,phys]x(n);[phys]bound(n,m=2);int [io,phys]maxit();int [io,phys]maxfc();
int [phys]nearlyq();int [phys]gtype();int [phys]htvtype();int [phys]trtype();int [phys]fmaxit();int [phys]gmaxit();
int [phys]interpmaxit();int [phys]cgstop();int [phys]cgmaxit();int [phys]qmpmaxit();
[phys]ftol();[phys]epsgpen();[phys]epsgpsn();[phys]cggtol();[phys]cgitol();[phys]cgftol();
[phys]qmptol();[phys]delta();[phys]eta();[phys]delmin();
int [phys]print(p);int [io,phys]info(); SV* f_func; SV* g_func; SV* h_func)
```

Solves the box-constrained minimization problem

```
Minimize f(x)
subject to l \leq x \leq u
```

using a method described in E. G. Birgin and J. M. Martinez, "Large-scale active-set box-constrained optimization method with spectral projected gradients", Computational Optimization and Applications 23, 101-125 (2002).

This is the simplified version of gencan. Subroutines evalf and evalg must be supplied by the user to evaluate the function f and its gradient, respectively. The calling sequences are

```
inform evalf(f, x)
inform evalg(g, x)
```

where x is the point where the function (the gradient) must be evaluated, n is the number of variables and f (g) is the functional value (the gradient vector). The real parameters x, f, g must be double precision.

A subroutine evalhd to compute the Hessian times vector products is optional. If this subroutine is not provided an incremental quotients version will be used instead. The calling sequence of this subroutine should be

```
inform call evalhd(hu, x, u, ind)
where x is the point where the approx-Hessian is being considered,
u is the vector which should be multiplied by the approx-Hessian H
and hu is the vector where the product should be placed.
```

The information about the matrix H must be passed to evalhd by means of common declarations. The necessary computations must be done in evalg. The real parameters x, u, hu must be double precision.

This subroutine must be coded by the user, taking into account that n is the number of variables of the problem and that hu must be the product H u. Moreover, you must assume, when you code evalhd, that only size(ind) components of u are nonnull and that ind is the set of indices of those components. In other words, you must write evalhd in such a way that hu is the vector whose i-th entry is

` hu(i) = \Sum_{j=1}^{nind} H_{i,ind(j)} u_ind(j)`

Moreover, the only components of hu that you need to compute are those which corresponds to the indices ind(1),...,ind(nind). However, observe that you must assume that, in u, the whole vector is present, with its n components, even the zeroes. So, if you decide to code evalhd without taking into account the presence of ind and nind, you can do it. A final observation: probably, if nind is close to n, it is not worthwhile to use ind, due to the cost of accessing the correct indices. If you want, you can test, within your evalhd, if (say) nind > n/2, and, in this case use a straightforward scalar product for the components of hu.

Example: Suppose that the matrix H is full. The main steps of evalhd could be:

```
do i= 1, nind
indi= ind(i)
hu(indi)= 0.0d0
do j= 1, nind
indj= ind(j)
hu(indi)= hu(indi) + H(indi,indj) * u(indj)
end do
end do
On Entry
x double precision x(n)
initial estimate to the solution
bounds(n,2)
lower bounds and upper bounds
epsgpen double precision
small positive number for declaring convergence when the
euclidian norm of the projected gradient is less than
or equal to epsgpen
RECOMMENDED: epsgpen = 1.0d-5
epsgpsn double precision
small positive number for declaring convergence when the
infinite norm of the projected gradient is less than
or equal to epsgpsn
RECOMMENDED: epsgpsn = 1.0d-5
ftol double precision
'lack of enough progress' measure. The algorithm stops by
'lack of enough progress' when f(x_k) - f(x_{k+1}) <=
ftol * max { f(x_j)-f(x_{j+1}, j<k} during fmaxit
consecutive iterations. This stopping criterion may be
inhibited setting ftol = 0. We recommend, preliminary, to
set ftol = 0.01 and fmaxit = 5
RECOMMENDED: ftol = 1.0d-2
fmaxit integer
see the meaning of ftol, above
RECOMMENDED: fmaxit = 5
gmaxit integer
If the order of the euclidian-norm of the continuous projected
gradient did not change during gmaxit consecutive iterations
the execution stops. Recommended: gmaxit= 10. In any case
gmaxit must be greater than or equal to 1
RECOMMENDED: gmaxit = 10
fmin double precision
function value for the stopping criteria f <= fmin
RECOMMENDED: fmin = -1.0d+99 (inhibited)
maxit integer
maximum number of iterations allowed
RECOMMENDED: maxit = 1000
maxfc integer
maximum number of funtion evaluations allowed
RECOMMENDED: maxfc = 5000
delta
initial trust-region radius. Default max{0.1||x||,0.1} is set
if you set delta < 0. Otherwise, the parameters delta
will be the ones set by the user.
RECOMMENDED: delta = -1
cgmaxit integer
maximum number of iterations allowed for the cg subalgorithm
Default values for this parameter and the previous one are
0.1 and 10 * log (number of free variables). Default values
are taken if you set ucgeps < 0 and cgmaxit < 0,
respectively. Otherwise, the parameters ucgeps and cgmaxit
will be the ones set by the user
RECOMMENDED: cgmaxit = -1
cgstop, cggtol double precision
cgstop means cunjugate gradient stopping criterion relation, and
cggtol means conjugate gradients projected gradient final norm.
Both are related to a stopping criterion of conjugate gradients.
This stopping criterion depends on the norm of the residual
of the linear system. The norm of the this residual should be
less or equal than a 'small'quantity which decreases as we are
approximating the solution of the minimization problem (near the
solution, better the truncated-Newton direction we aim). Then, the
log of the required precision requested to conjugate gradient has
a linear dependence on the log of the norm of the projected
gradient. This linear relation uses the squared euclidian-norm
of the projected gradient if cgstop = 1 and uses the sup-norm if
cgstop = 2. In adition, the precision required to CG is equal to
cgitol (conjugate gradient initial epsilon) at x0 and cgftol
(conjugate gradient final epsilon) when the euclidian- or sup-norm
of the projected gradient is equal to cggtol (conjugate gradients
projected gradient final norm) which is an estimation of the value
of the euclidian- or sup-norm of the projected gradient at the
solution.
RECOMMENDED: cgstop = 1, cggtol = epsgpen; or
cgstop = 2, cggtol = epsgpsn.
cgitol, cgftol double precision
small positive numbers for declaring convergence of the
conjugate gradient subalgorithm when
||r||_2 < cgeps * ||rhs||_2, where r is the residual and
rhs is the right hand side of the linear system, i.e., cg
stops when the relative error of the solution is smaller
that cgeps.
cgeps varies from cgitol to cgftol in such a way that, depending
on cgstop (see above),
i) log10(cgeps^2) depends linearly on log10(||g_P(x)||_2^2)
which varies from ||g_P(x_0)||_2^2 to epsgpen^2; or
ii) log10(cgeps) depends linearly on log10(||g_P(x)||_inf)
which varies from ||g_P(x_0)||_inf to epsgpsn.
RECOMMENDED: cgitol = 1.0d-1, cgftol = 1.0d-5
qmptol double precision
see below
qmpmaxit integer
This and the previous one parameter are used for a stopping
criterion of the conjugate gradient subalgorithm. If the
progress in the quadratic model is less or equal than a
fraction of the best progress ( qmptol * bestprog ) during
qmpmaxit consecutive iterations then CG is stopped by not
enough progress of the quadratic model.
RECOMMENDED: qmptol = 1.0d-4, qmpmaxit = 5
nearlyq logical
if function f is (nearly) quadratic, use the option
nearlyq = 0 Otherwise, keep the default option.
if, at an iteration of CG we find a direction d such
that d^T H d <= 0 then we take the following decision:
(i) if nearlyq = 1 then take direction d and try to
go to the boundary chosing the best point among the two
point at the boundary and the current point.
(ii) if nearlyq = 0 then we stop at the current point.
RECOMMENDED: nearlyq = 0
gtype integer
type of gradient calculation
gtype = 0 means user suplied evalg subroutine,
gtype = 1 means central diference approximation.
RECOMMENDED: gtype = 0
(provided you have the evalg subroutine)
htvtype integer
type of gradient calculation
htvtype = 0 means user suplied evalhd subroutine,
htvtype = 1 means incremental quotients approximation.
RECOMMENDED: htvtype = 1
(you take some risk using this option but, unless you have
a good evalhd subroutine, incremental quotients is a very
cheap option)
trtype integer
type of trust-region radius
trtype = 0 means 2-norm trust-region
trtype = 1 means infinite-norm trust-region
RECOMMENDED: trtype = 0
print(0) integer
commands printing. Nothing is printed if print < 0.
If print = 0, only initial and final information is printed.
If print > 0, information is printed every print iterations.
Exhaustive printing when print > 0 is commanded by print(1).
RECOMMENDED: print(0) = 1
print(1) integer
When print(0) > 0, detailed printing can be required setting
print(1) = 1.
RECOMMENDED: print(1) = 1
eta double precision
constant for deciding abandon the current face or not
We abandon the current face if the norm of the internal
gradient (here, internal components of the continuous
projected gradient) is smaller than (1-eta) times the
norm of the continuous projected gradient. Using eta=0.9
is a rather conservative strategy in the sense that
internal iterations are preferred over SPG iterations.
RECOMMENDED: eta = 0.9
delmin double precision
minimum 'trust region' to compute the Truncated Newton
direction
RECOMMENDED: delmin = 0.1
interpmaxit integer
constant for testing if, after having made at least interpmaxit
interpolations, the steplength is too small. In that case
failure of the line search is declared (may be the direction
is not a descent direction due to an error in the gradient
calculations)
RECOMMENDED: interpmaxit = 4
(use interpmaxit > maxfc for inhibit this stopping criterion)
On Return
x double precision x(n)
final estimation to the solution
f double precision
function value at the final estimation
g double precision g(n)
gradient at the final estimation
maxit
number of iterations
maxfc
number of function evaluations
info
This output parameter tells what happened in this
subroutine, according to the following conventions:
0= convergence with small euclidian-norm of the
projected gradient (smaller than epsgpen);
1= convergence with small infinite-norm of the
projected gradient (smaller than epsgpsn);
2= the algorithm stopped by 'lack of enough progress',
that means that f(x_k) - f(x_{k+1}) <=
ftol * max { f(x_j)-f(x_{j+1}, j<k} during fmaxit
consecutive iterations;
3= the algorithm stopped because the order of the euclidian-
norm of the continuous projected gradient did not change
during gmaxit consecutive iterations. Probably, we
are asking for an exagerately small norm of continuous
projected gradient for declaring convergence;
4= the algorithm stopped because the functional value
is very small (f <= fmin);
6= too small step in a line search. After having made at
least interpmaxit interpolations, the steplength becames
small. 'small steplength' means that we are at point
x with direction d and step alpha, and
alpha * ||d||_infty < max(epsabs, epsrel * ||x||_infty).
In that case failure of the line search is declared
(may be the direction is not a descent direction
due to an error in the gradient calculations). Use
interpmaxit > maxfc for inhibit this criterion;
7= it was achieved the maximum allowed number of
iterations (maxit);
8= it was achieved the maximum allowed number of
function evaluations (maxfc);
9= error in evalf subroutine;
10= error in evalg subroutine;
11= error in evalhd subroutine.
```

```
$x = random(5);
$gx = $x->zeroes;
$fx = pdl(0);
$print = pdl(long,[1,0]);
$maxit = pdl(long, 200);
$maxfc = pdl(long, 1000);
$info = pdl(long,0);
$bounds = zeroes(5,2);
$bounds(,0).=-5;
$bounds(,1).=5;
sub f_func{
my ($fx, $x) = @_;
$fx .= rosen($x);
return 0;
}
sub g_func{
my ($gx, $x) = @_;
$gx .= rosen_grad($x);
return 0;
}
sub h_func{
my ($hx, $x, $d, $ind) = @_;
$hx .= rosen_hess($x,1) x $d;
return 0;
}
sgencan($fx, $gx, $x, $bounds, $maxit, $maxfc,
1, 0, 0, 0, 5, 10, 5, 1, -1, 5,
0, 1e-8, 1e-10, 1e-5, 0.1, 1e-5, 1e-5,
-1, 0.9, 0.1,
$print,$info,\&f_func,\&g_func, \&h_func);
```

sgencan ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## dhc

` Signature: ([io,phys]xrandom(n,m);step(); xtol(); int print();[o]fx();[o]x(n); SV* dhc_func)`

Find a point X where the function dhc_func(X) has a global minimum. X is an n-vector and f(X) is a scalar. In mathematical notation

` f: R^n -> R^1.`

using a method, dynamic hill climbing, described in D. Yuret, "From Genetic Algorithms To Efficient Optimization", A.I. Technical Report No. 1569 (1994). (http://home.ku.edu.tr/~dyuret/pub/aitr1569.html).

```
where
fx: On exit it contains the value of the function f at
the point x.
x: On exit this is the location of the global minimum,
calculated by the program.
xrandom: This is a user-supplied initial starting locations.
On exit there are locations of the minimums
calculated by the program.
step: Initial step length.
xtol: Step tolerance(minimum step size).
print: if true print some information at each iteration
(each minimum).
dhc_func: Objective function to be minimized.
If you need boundary conditions, put them in the
objective function such that the optimizer gets
bad values for points out of bounds.
scalar double dhc_func($x())
```

```
# Local Optimization
$randomx = grandom(2);
sub test{
my $a = shift;
rosen($a)->sclr;
}
$step = pdl(1.0);
$tol = pdl(1e-10);
($fx , $ret) = dhc($randomx, $step, $tol,0,\&test);
print "Minimum found ($fx) at $ret";
# Try to solve
# The SIAM 100-Digit Challenge problem 4
# see http://www-m8.ma.tum.de/m3/bornemann/challengebook/
# result: -3.30686864747523728007611377089851565716648236
$randomx = (random(2,100)-0.5)*2;
sub test{
my $x = shift;
my $f = exp(sin(50*$x(0)))+sin(60*exp($x(1)))+
sin(70*sin($x(0)))+sin(sin(80*$x(1)))-
sin(10*($x(0)+$x(1)))+($x(0)**2+$x(1)**2)/4;
$f->sclr;
}
$step = pdl(0.7);
$tol = pdl(1e-8);
($fx , $ret) = dhc($randomx, $step, $tol,0,\&test);
print "Minimum found ($fx) at $ret";
```

dhc ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## de_opt

```
Signature: ([io,phys]x(n);int genmax();int seed();int strategy();
int np();f();cr();inibound_l();inibound_u();int print();[o]fx();[o]cvar(); SV* de_func)
```

Find a point X where the function de_func(X) has a global minimum. X is an n-vector and f(X) is a scalar. In mathematical notation

` f: R^n -> R^1.`

using a method described in Storn, R. and Price, K., "Differential Evolution - a Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces", Technical Report TR-95-012, ICSI, March 1995. (http://www.icsi.berkeley.edu/~storn/code.html)

Strategy:

```
1: "DE/best/1/exp",
2: "DE/rand/1/exp",
3: "DE/rand-to-best/1/exp",
4: "DE/best/2/exp",
5: "DE/rand/2/exp",
6: "DE/best/1/bin",
7: "DE/rand/1/bin",
8: "DE/rand-to-best/1/bin",
9: "DE/best/2/bin",
10: "DE/rand/2/bin"
Choice of strategy
We have tried to come up with a sensible naming-convention: DE/x/y/z
DE : stands for Differential Evolution
x : a string which denotes the vector to be perturbed
y : number of difference vectors taken for perturbation of x
z : crossover method (exp = exponential, bin = binomial)
There are some simple rules which are worth following:
F is usually between 0.5 and 1 (in rare cases > 1
CR is between 0 and 1 with 0., 0.3, 0.7 and 1. being worth to be tried first
To start off NP = 10*D is a reasonable choice. Increase NP if misconvergence happens.
If you increase NP, F usually has to be decreased
When the DE/best... schemes fail DE/rand... usually works and vice versa
where
fx: On exit it contains the value of the function f at
the point x.
x: On exit this is the location of the global minimum,
calculated by the program.
cvar: On exit it contains the value variance of the function f.
strategy: Choice of strategy.
seed: Random seed.
genmax: Maximum number of generations.
np: Population size.
cr: Crossing over factor.
f: Weight factor.
inibound_l: Lower parameter bound for init.
inibound_u: Upper parameter bound for init.
print: if > 1 print some information at each 'print' generation
(minimum = 1).
de_func: Objective function to be minimized.
If you need boundary conditions, put them in the
objective function such that the optimizer gets
bad values for points out of bounds.
scalar double de_func($x())
```

```
# Try to solve
# The SIAM 100-Digit Challenge problem 4
# see http://www-m8.ma.tum.de/m3/bornemann/challengebook/
# result: -3.30686864747523728007611377089851565716648236
use PDL::Opt::NonLinear;
$x = zeroes(2);
$strategy = pdl(long,7);
$np = pdl(long,50);
$print = pdl(long,50);
$inibound_l = pdl(-1.0);
$inibound_h = pdl(1.0);
$genmax = pdl(long,250);
$seed = pdl(long,3);
$f = pdl(0.9);
$cr = pdl(0.9);
sub test{
my $x = shift;
my ($x0, $y1);
$x0 = PDL::Core::sclr_c($x(0));
$y1 = PDL::Core::sclr_c($x(1));
my $f = exp(sin(50*$x0))+sin(60*exp($y1))+
sin(70*sin($x0))+sin(sin(80*$y1))-
sin(10*($x0+$y1))+($x0**2+$y1**2)/4;
$f;
}
($fx,$cvar)=de_opt($x, $genmax, $seed, $strategy, $np,
$f, $cr, $inibound_l, $inibound_h,
$print, \&test);
print "Minimum found ($fx) at $x with variance $cvar\n";
```

de_opt ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## asa_opt

```
Signature: ([io,phys]x(n);int seed();inibound_l(n);inibound_u(n);
int parameter_type(n); int limit(5); cost_param(4); temperature(3);
int generic(10);resolution(n);coarse_resolution(n);
quench_cost(); quench_param(n);int print();[o]fx();[o]tangents(n);[o]curvature(n,n);int [o]info(); SV* asa_func)
```

This routine solves the optimization problem

```
minimize f(x)
x
subject to low <= x <= up
```

It uses the Adaptive Simulated Annealing (ASA) method. (see http://www.ingber.com/#ASA-CODE)

```
where
x
is a double precision array of dimension n.
On entry x is an approximation to the solution.
On exit x is the current approximation.
seed
random seed.
inibound_l
lower bound on x.
inibound_u
upper bound on x.
parameter_type
type of value of x
-2 => real value, no reanneal
-1 => real value
1 => integral value
2 => integral value, no reanneal
limit
limit(0) = Maximum_Cost_Repeat
limit(1) = Number_Cost_Samples
limit(2) = Limit_Acceptances
limit(3) = Limit_Generated
limit(4) = Limit_Invalid_Generated_States
cost_param
cost_param(0) = Accepted_To_Generated_Ratio
cost_param(1) = Cost_Precision
cost_parma(2) = Cost_Parameter_Scale_Ratio
cost_parma(3) = Delta_X
temperature
temperature(0) = Initial_Parameter_Temperature
temperature(1) = Temperature_Ratio_Scale
temperature(2) = Temperature_Anneal_Scale
generic
generic(0) = Include_Integer_Parameters
generic(1) = User_Initial_Parameters
generic(2) = Sequential_Parameters
generic(3) = Acceptance_Frequency_Modulus
generic(4) = Generated_Frequency_Modulus
generic(5) = Reanneal_Cost
generic(6) = Reanneal_Parameters
generic(7) = Queue_Size
generic(8) = User_Tangents (not implemented)
generic(9) = Curvature_0
resolution
On entry, array of resolutions used to compare
the currently generated parameters to those in the queue.
coarse_resolution
On entry, array of resolutions used to resolve
the values of generated parameters.
quench_cost
used to adaptively set the scale of the temperature schedule.
quench_param
used to adaptively set the scale of the temperature schedule.
print
print = 0 no output is generated
fx
On final exit f is the value of the function at x.
tangents
On exit, it is the value of the tangents (gradient) at x.
curvature
On exit, it is the value of the curvature (hessian) at x.
info
On entry 0,
On exit, contain error code:
NORMAL_EXIT => 0
P_TEMP_TOO_SMALL => 1
C_TEMP_TOO_SMALL => 2
COST_REPEATING => 3
TOO_MANY_INVALID_STATES => 4
IMMEDIATE_EXIT => 5
INVALID_USER_INPUT => 7
INVALID_COST_FUNCTION => 8
INVALID_COST_FUNCTION_DERIV => 9
CALLOC_FAILED => -1
```

```
# Try to solve
# The SIAM 100-Digit Challenge problem 4
# see http://www-m8.ma.tum.de/m3/bornemann/challengebook/
# result: -3.30686864747523728007611377089851565716648236
use PDL::Opt::NonLinear;
sub test{
my $x = shift;
my ($x0, $y1);
$x0 = PDL::Core::sclr_c($x(0));
$y1 = PDL::Core::sclr_c($x(1));
my $f = exp(sin(50*$x0))+sin(60*exp($y1))+
sin(70*sin($x0))+sin(sin(80*$y1))-
sin(10*($x0+$y1))+($x0**2+$y1**2)/4;
$f;
}
$bu = zeroes(2);
$bl = zeroes(2);
$bu .= 1;
$bl .= -1;
$seed = pdl(696969);
$parameter = zeroes(long,2);
$parameter .= -1;
$qp = ones(2);
$qc = pdl(1.0);
$print = pdl(long,0);
$seed = pdl(long, 696969);
$limit = pdl(long,[5,10,1000,99999,1000]);
$cost_param = pdl [1.e-4,1.e-18,1.0,0.001];
# $temp = pdl [1.0,1.e-5,100.0]; for generic problem
$temp = pdl [1.0,1.e-5,10000.0];
$generic = pdl(long,[0,0,-1,100,10000,1,1,50,0,0]);
$res = zeroes(2);
$coarse = zeroes(2);
$x = (random(2)-0.5)*2;
asa_opt($x, ++$seed, $bl, $bu, $parameter, $limit, $cost_param, $temp,
$generic, $res, $coarse, $qc, $qp, $print, \&test);
# Local optimize now
$rho = pdl(0.2);
$tol = pdl(1e-10);
$maxit =pdl(long, 500);
$x->hooke($maxit, $rho,$tol,\&test);
print "Minimum found ".test($x)." at $x in $maxit iteration(s)";
```

asa_opt ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

# COPYRIGHT

Copyright (C) Grégory Vanuxem 2005-2018. All rights reserved. This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.

1 POD Error

The following errors were encountered while parsing the POD:

- Around line 223:
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