++ed by:

# 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);
}
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;
}
($aa,$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
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
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
||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
gtype = 0 means user suplied evalg subroutine,
gtype = 1 means central diference approximation.

RECOMMENDED: gtype = 0

(provided you have the evalg subroutine)

htvtype integer
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
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)

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

1= convergence with small infinite-norm of the

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

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 (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

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