Here is a short wrapper which automates repeated calls to the DirectSearch 2 curve-fitting routine. It offers both time and repetition (solver restart) limits.

The global optimization package DirectSearch 2 (see Application Center link, and here) has some very nice features. One aspect which I really like is that it can do curve-fitting: to fit an expression using tabular data. By this, I mean that it can find optimal values of parameters present in an expression (formula) such that the residual error between that formula and the tabular data is minimized.

Maple itself has commands from the CurveFitting and Statistics packages for data regression, such as NonlinearFit, etc. But those use local optimization solvers, and quite often for the nonlinear case one may need a global optimizer in order to produce a good fit. The nonlinear problem may have local extrema which are not even close to being globally optimal or provide a close fit.

Maplesoft offers the (commercially available) GlobalOptimization package as an add-on to Maple, but its solvers are not hooked into those mentioned curve-fitting commands. One has to set up the proper residual-based objective function onself in order to use this for curve-fitting, and some of the bells and whistles may be harder to do.

So this is why I really like the fact that the DirectSearch 2 package has its own exported commands to do curve-fitting, integrated with its global solvers.

But as the DirectSearch package's author mentions, the fitting routine may sometimes exit too early. Repeat starts of the solver, for the very same parameter ranges, can produce varying results due to randomization steps performed by the solver. This post is branched off from another thread which involved such a problematic example.

Global optimization is often a dark art. Sometimes one may wish to simply have the engine work for 24 hours, and produce whatever best result it can. That's the basic enhancement this wrapper offers.

Here is the wrapper, and a few illustrative calls to it on the mentioned curve-fitting example that show informative progress status messages, etc. I've tried to make the wrapper pretty generic. It could be reused for other similar purposes.

Other improvements are possible, but might make it less generic. A target option is possible, where attainment of the target would cause an immediate stop. The wrapper could be made into an appliable module, and the running best result could be stored in a module local so that any error (and ensuing halt) would not wipe out the best result from potentially hours and hours worth of conputation.

restart:
randomize():
repeater:=proc( funccall::uneval
, {maxtime::numeric:=60}
, {maxiter::posint:=10}
, {access::appliable:=proc(a) SFloat(a[1]); end proc}
, {initial::anything:=[infinity]}
)
local best, current, elapsed, i, starttime;
starttime:=time[real]();
elapsed:=time[real]()-starttime;
i:=1; best:=[infinity];
while elapsed<maxtime and i<=maxiter do
userinfo(2,repeater,`iteration `,i);
try
timelimit(maxtime-elapsed,assign('current',eval(funccall)));
catch "time expired":
end try;
if is(access(current)<access(best)) then
best:=current;
userinfo(1,repeater,`new best `,access(best));
end if;
i:=i+1;
elapsed:=time[real]()-starttime;
userinfo(2,repeater,`elapsed time `,elapsed);
end do;
if best<>initial then
return best;
else
error "time limit exceeded during first attempt";
end if;
end proc:
X := Vector([seq(.1*j, j = 0 .. 16), 1.65], datatype = float):
Y := Vector([2.61, 2.62, 2.62, 2.62, 2.63, 2.63, 2.74, 2.98, 3.66,
5.04, 7.52, 10.74, 12.62, 10.17, 5, 2.64, 11.5, 35.4],
datatype = float):
F := a*cosh(b*x^c*sin(d*x^e));
/ c / e\\
F := a cosh\b x sin\d x //
infolevel[repeater]:=2: # or 1, or not at all (ie. 0)
interface(warnlevel=0): # disabling warnings. disable if you want.
repeater(DirectSearch:-DataFit(F
, [a=0..10, b=-10..10, c=0..100, d=0..7, e=0..4]
, X, Y, x
, strategy=globalsearch
, evaluationlimit=30000
));
repeater: iteration 1
repeater: new best 9.81701944539358706
repeater: elapsed time 15.884
repeater: iteration 2
repeater: new best 2.30718902535293857
repeater: elapsed time 22.354
repeater: iteration 3
repeater: new best 0.627585701120743822e-4
repeater: elapsed time 30.777
repeater: iteration 4
repeater: elapsed time 47.959
repeater: iteration 5
repeater: new best 0.627585700905294148e-4
repeater: elapsed time 55.221
repeater: iteration 6
repeater: elapsed time 60.009
[0.0000627585700905294, [a = 2.61748237902808, b = 1.71949329097179,
c = 2.30924401405164, d = 1.50333106110324, e = 1.84597267458055], 4333]
# without userinfo messages printed
infolevel[repeater]:=0:
repeater(DirectSearch:-DataFit(F
, [a=0..10, b=-10..10, c=0..100, d=0..7, e=0..4]
, X, Y, x
, strategy=globalsearch
, evaluationlimit=30000
));
[0.0000627585701341043, [a = 2.61748226209478, b = 1.71949332125427,
c = 2.30924369227236, d = 1.50333090706676, e = 1.84597294290477], 6050]
# illustrating early timeout
infolevel[repeater]:=2:
repeater(DirectSearch:-DataFit(F
, [a=0..10, b=-10..10, c=0..100, d=0..7, e=0..4]
, X, Y, x
, strategy=globalsearch
, evaluationlimit=30000
),
maxtime=2);
repeater: iteration 1
repeater: elapsed time 2.002
Error, (in repeater) time limit exceeded during first attempt
# illustrating iteration limit cutoff
infolevel[repeater]:=2:
repeater(DirectSearch:-DataFit(F
, [a=0..10, b=-10..10, c=0..100, d=0..7, e=0..4]
, X, Y, x
, strategy=globalsearch
, evaluationlimit=30000
),
maxiter=1);
repeater: iteration 1
repeater: new best 5.68594272127419575
repeater: elapsed time 7.084
[5.68594272127420, [a = 3.51723075672918, b = -1.48456068506828,
c = 1.60544055207338, d = 6.99999999983179, e = 3.72070034285212], 2793]
# giving it a large total time limit, with reduced userinfo messages
infolevel[repeater]:=1:
Digits:=15:
repeater(DirectSearch:-DataFit(F
, [a=0..10, b=-10..10, c=0..100, d=0..7, e=0..4]
, X, Y, x
, strategy=globalsearch
, evaluationlimit=30000
),
maxtime=2000, maxiter=1000);
repeater: new best 3.10971990123465947
repeater: new best 0.627585701270853103e-4
repeater: new best 0.627585700896181428e-4
repeater: new best 0.627585700896051324e-4
repeater: new best 0.627585700895833535e-4
repeater: new best 0.627585700895607885e-4
[0.0000627585700895608, [a = 2.61748239185387, b = -1.71949328487160,
c = 2.30924398692221, d = 1.50333104262348, e = 1.84597270535142], 6502]