Global Optimization Toolbox sensitive to search ranges?
Hi
I'm trying to work out how the Global Optimization Toolbox works and I thought I got it working. However, when I increase the ranges of the parameters over which the Toolbox should look for the solution (but still including the previous range), the outcome of the optimization changes, i.e. the toolbox finds different optimal parameter values and a different, and often higher, Sum of Squares (which is my evaluation criterium). I don't understand what I'm doing wrong!
An example:
if I run the program with VarRanges: [r=0..10,a=0..100,k=0.0..10,s=0..0.1], the parameter values found are a=0, k=0.004, r=0.006, s=0.018 and the Sum of Squares=30.87
if I run the program with a broader range for s (s=0..1), the parameter values found are a=2.84, k=0.95, r=0.0026, s=0.80 and the Sum of Squares is much higher than found before (128.75)
I don't understand because the better solution found with the first VarRanges is included in the ranges set in the second part of the example. Still, the toolbox does not find that solution.
I've pasted my worksheet below. I hope somebody can help me!
Thanks, Marjet
restart:
with(GlobalOptimization):
Stime:=[0., 2.0, 4.0, 8.0, 12.0, 20.0]:
Sdata:=[5.703782474656201, 16.588099280204055, 15.201804919084164, 14.77102200299171, 16.300417207752275, 18.03501826314038]:
Tspline:=piecewise(v < 2.0, 859.4940801+614.505919899999980*v-242.505919900000009*(v-1.0)^3, v < 3.0, 2072.023679-113.011839699999996*v-727.517759814391980*(v-2.0)^2+374.529599700000006*(v-2.0)^3, v < 4.0, 2713.375682-444.458560499999976*v+396.071039257567974*(v-3.0)^2-117.612478800000006*(v-3.0)^3, v < 8.0, 1234.615674-5.15391840000000024*v+43.2336027841202011*(v-4.0)^2-4.39253079700000004*(v-4.0)^3, v < 12.0, 565.012594+129.873425700000012*v-9.47676677469249462*(v-8.0)^2-3.84164741000000020*(v-8.0)^3, v < 16.0, 3290.077410-130.339784199999997*v-55.5765356853502226*(v-12.0)^2+9.79037043299999964*(v-12.0)^3, v < 20.0, 2622.228621-105.014288800000002*v+61.9079095160933904*(v-16.0)^2-8.75733432700000058*(v-16.0)^3, 1554.061206-30.1030602999999992*v-43.1801023790233529*(v-20.0)^2+3.59834186699999980*(v-20.0)^3):
Espline:=piecewise(v < 2.0, 1353.868169-37.8681685999999972*v+85.8681686000000042*(v-1.0)^3, v < 3.0, 924.5273256+219.736337200000008*v+257.604505781836906*(v-2.0)^2-66.3408429699999971*(v-2.0)^3, v < 4.0, 167.231540+535.922819900000036*v+58.5819768726522838*(v-3.0)^2-63.5047967300000026*(v-3.0)^3, v < 8.0, 455.710466+462.572383500000001*v-131.932413272446070*(v-4.0)^2+13.8223293599999996*(v-4.0)^3, v < 12.0, 2365.320910+70.5848862100000076*v+33.9355389629520516*(v-8.0)^2-9.22356512700000053*(v-8.0)^3, v < 16.0, 4372.943140-100.661928300000000*v-76.7472425793621512*(v-12.0)^2+22.0094311699999992*(v-12.0)^3, v < 20.0, -2526.005227+341.812826700000016*v+187.365931354496582*(v-16.0)^2-37.1266595299999978*(v-16.0)^3, 3758.787586+58.6606207000000026*v-258.153982838624130*(v-20.0)^2+21.5128318999999984*(v-20.0)^3):
f:=(t)->(5.7 + r*(((evalf(int(Tspline,v=0..t)))+(evalf(int(abs(Tspline),v=0..t))))/2) - a*t - k*(((evalf(int(((Espline)*(max((1-s*t),0))),v=0..t)))+(evalf(int(((abs(Espline))*(max((1-s*t),0))),v=0..t))))/2) ):
Difference :=[seq((f(convert(Stime[p],float))-Sdata[p], p=1..6))]:
VarRanges:=[r=0..10,a=0..100,k=0.0..10,s=0..1]:
GlobalSolve(add(Difference[p]^2,p=1..6), op(VarRanges));
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method=branchandbound
Add method=branchandbound to the GlobalSolve command, i.e.
GlobalSolve(add(Difference[p]^2,p=1..6), op(VarRanges), method=branchandbound);
It's now insensitive to your search range. The default method is multistart.
Samir
Global optimization and search ranges
The solver in the Global Optimization Toolbox is most certainly dependent on the search ranges, and this applies to all the methods available. The local optimization solvers in the Optimization package generally start from an initial point and use derivative information to search for a local minimum from that point. However, the global solvers partition the search space specified by the ranges, and so providing different ranges can give different results.
Keep in mind the limitations of a global optimization solver based on numerical methods. It's easy to think that the solver should return the true optimal solution, but in reality, the solver returns the best solution achieved before the stopping criteria are met. The stopping criteria include such things as an optimality tolerance and a computational time limit. Changing the ranges will change how the search proceeds and possibly which stopping criteria are met.
Also note that the methods used by the GlobalOptimization solver do make some continuity assumptions. Most of the time, the solver returns a reasonable answer even if these assumptions are not met, but there's no guarantee the true optimum will be found.
The GlobalOptimization/Computation help page should give more details about these matters. You could try setting infolevel[GlobalOptimization] to a value greater than 1 to view more information about the stopping criteria used during the computation.
Paulina Chin
Maplesoft