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Re: Re: FindMinimum and the minimum-radius circle
For both data sets, I can plainly SEE that the answer is wrong on a plot, as the circle clearly doesn't contain three of the data points and clearly doesn't contain two of them on a diameter. I'm not talking about sixth-decimal-place errors.
The other methods that survived this far don't make visible errors, and Fred Simons' method is also faster.
Bobby
On Sat, 28 Aug 2004 04:38:01 -0400 (EDT), Maxim <ab_def at prontomail.com> wrote:
> I just have a small objection to using the word 'fails'; after all,
> we're talking about numerical optimization. For example, here's what
> NMinimize gives for my first example:
>
> In[1]:=
> SeedRandom[1];
> points = Partition[Table[Random[], {1000}], 2];
>
> goal[x_, y_] = Max[Norm[{x, y} - #]& /@ points];
>
> grad = Compile[
> {x, y, {points, _Real, 2}},
> Module[
> {n = Length@points, Ld = Norm[{x, y} - #]& /@ points,
> Ldd, i1, i2},
> {i2, i1} = Ordering[Ld, -2];
> Ldd = ({x, y} - points[[#]])/Ld[[#]]& /@ {i1, i2};
> If[Chop[Ld[[i1]] - Ld[[i2]]] != 0,
> Ldd[[1]],
> (Ldd[[1]] + Ldd[[2]])/2
> ]],
> {{_Ordering, _Integer, 1}}
> ];
>
> NMinimize[goal[x, y], {x, y}]
>
> Out[5]=
> {0.66281538, {x -> 0.48683054, y -> 0.4614583}}
>
> It turns out that this is not the exact minimum either. Does it mean
> that NMinimize fails as well? The advantage of the FindMinimum
> approach is that FindMinimum can be called repeatedly to obtain finer
> approximations:
>
> In[6]:=
> FixedPoint[
> FindMinimum[goal[x, y],
> {x, x /. #[[2]]}, {y, y /. #[[2]]},
> Method -> Gradient, Gradient :> grad[x, y, points]]&,
> {Infinity, {x -> Mean[points][[1]], y -> Mean[points][[2]]}}]
>
> Out[6]=
> {0.66281504, {x -> 0.48684176, y -> 0.46146944}}
>
> If you want speed increase, replace Norm in the definitions of goal
> and grad with Sqrt[#.#]&; Norm is slow. Certainly some more
> fine-tuning can be done; for example, replace Chop[Ld[[i1]]-Ld[[i2]]]
> with Chop[Ld[[i1]]-Ld[[i2]],10^-6], this seems to give better results
> in most cases.
>
> Maxim Rytin
> m.r at inbox.ru
>
>
> DrBob <drbob at bigfoot.com> wrote in message news:<cgmmqi$ts$1 at smc.vnet.net>...
>> That frequently works, and it's almost as fast as the Fred Simons method.
>>
>> But it fails on this data:
>>
>> {{-0.694112,-1.47331},{-2.26344,0.220021},{-0.8867,-1.85288},{0.527656,-2.\
>> 81915},{-1.75879,-0.28898},{
>> 0.728263,1.20724},{-1.34788,-0.799066},{-0.676629,-1.03418},{1.45111,0.\
>> 961224},{0.914209,-0.539339}}
>>
>> and this data:
>>
>> {{1.60034, 0.0176019}, {-0.355331, -1.62872}, {-0.2654, -0.000635396}, \
>> {-0.777536, 0.890349}}
>>
>> In both cases, the result circle is too big and passes through two points when it should be three.
>>
>> I'm using Mean@data as the starting point (as you did), but I'm using the ConvexHull to define goal[x_,y_] and as the third argument of grad.
>>
>> Bobby
>>
>> On Thu, 26 Aug 2004 06:51:34 -0400 (EDT), Maxim <ab_def at prontomail.com> wrote:
>>
>> > Daniel Lichtblau <danl at wolfram.com> wrote in message news:<cg97e9$a5p$1 at smc.vnet.net>...
>> >> Janos D. Pinter wrote:
>> >> > Hello mathgroup,
>> >> >
>> >> > as stated in previous messages, the minimum-radius circle problem is
>> >> > convex, but constrained: hence, an unconstrained optimization approach may
>> >> > or may not work...
>> >>
>> >> But it can certainly be made to work on an unconstrained formulation.
>> >> This takes some nondefault option settings to work around problems that
>> >> arise because such formulation is not everywhere smooth.
>> >>
>> >>
>> >> > Using the convex hull as a constraint set will work, of
>> >> > course. NMInimize and other constrained optimizers should then also work,
>> >> > even in local search mode if they have that option.
>> >> >
>> >> > Instead of further philosophical discussions on the subject, why don't we
>> >> > solve a standardized test model(s), using the same numerical example. (I
>> >> > assume here that SeedRandom is platform and version independent... as it
>> >> > should be.)
>> >> >
>> >> > Here is a simple model-instance, for your perusal and experiments.
>> >> >
>> >> > SeedRandom[1];
>> >> > points = Partition[Table[Random[], {1000}], 2];
>> >> > constraints = (#[[1]] - x0)^2 + (#[[2]] - y0)^2 <= r^2 & /@ points;
>> >> > soln = NMinimize[{r, constraints}, {x0, y0, {r, 0, 10}}]
>> >> > {0.6628150350002375, {r -> 0.6628150350002375, x0 -> 0.48684175822621106,
>> >> > y0 -> 0.46146943870460916}}
>> >> >
>> >> > timedsoln = NMinimize[{r, constraints}, {x0, y0, {r, 0, 10}}] // AbsoluteTiming
>> >> > {30.694136`8.938600406559628*Second, {0.6628150350002375,
>> >> > {r -> 0.6628150350002375, x0 -> 0.48684175822621106, y0 ->
>> >> > 0.46146943870460916}}}
>> >> >
>> >> > That is (on a P4 1.6 WinXP machine) it takes about 30 seconds to generate
>> >> > this solution.
>> >> >
>> >> > Next, I solve the same model using the MathOptimizer Professional package,
>> >> > in local search mode:
>> >> >
>> >> > Needs["MathOptimizerPro`callLGO`"]
>> >> > callLGO[r, constraints, {{x0, 0, 1}, {y0, 0, 1}, {r, 0, 1}}, Method -> LS]
>> >> > // AbsoluteTiming
>> >> > {6.158856`8.24104504348061*Second,
>> >> > {0.662815036, {x0 -> 0.4868417589, y0 -> 0.4614694394, r -> 0.662815036},
>> >> > 4.770200900949817*^-11}}
>> >> >
>> >> > The two solutions are fairly close (the solution methods are different).
>> >> > The MathOptimizer Professional (LGO) solver time is ~ 6 seconds. The result
>> >> > also shows that the max. constraint error of this solution is ~
>> >> > 4.770200900949817*^-11. (This error could be reduced, if necessary by
>> >> > changing the model and some LGO options.)
>> >> >
>> >> > Obviously, one could use also more sophisticated models and other point
>> >> > sets etc., as long as we all use the same one(s), for objective
>> >> > comparisons. (The proof of the pudding principle.)
>> >> >
>> >> > Regards,
>> >> > Janos
>> >> > [...]
>> >>
>> >> What about the unconstrained formulation?
>> >>
>> >> In[47]:=
>> >> Timing[NMinimize[Sqrt[Max[Map[(x-#[[1]])^2+(y-#[[2]])^2&,points]]],{x,y}]]
>> >>
>> >> Out[47]=
>> >> {3.54 Second,{0.662815,{x\[Rule]0.486831,y\[Rule]0.461458}}}
>> >>
>> >> I note that this can be made faster still by preprocessing to extract
>> >> the convex hull of the data points. Also one can attain a faster result,
>> >> with but slight loss of accuracy, using
>> >>
>> >> FindMinimum[...,Gradient->{"FiniteDifference","DifferenceOrder" -> 2}]
>> >>
>> >> You can do this with arbitrary reasonable starting points but clearly
>> >> the mean or median values would be good choices.
>> >>
>> >> In[52]:=
>> >> Timing[FindMinimum[
>> >> Sqrt[Max[
>> >> Map[(x-#[[1]])^2+(y-#[[2]])^2&,points]]],{x,#[[1]]},{y,#[[2]]},
>> >> Gradient->{"FiniteDifference","DifferenceOrder"->2}]&[
>> >> Mean[points]]]
>> >>
>> >> Out[52]=
>> >> {1.34 Second,{0.663207,{x->0.487394,y->0.461453}}}
>> >>
>> >> Experimentation reveals that by taking DifferenceOrder->4 I can recover
>> >> the NMinimize/MOP result for this particular example, albeit at some
>> >> loss of speed.
>> >>
>> >> Again, this will perform much better if you work with the convex hull.
>> >> Here is how fast this becomes.
>> >>
>> >> Needs["DiscreteMath`ComputationalGeometry`"]
>> >>
>> >> In[69]:=
>> >> Timing[With[{hull=points[[ConvexHull[points]]]},
>> >> FindMinimum[
>> >> Sqrt[Max[
>> >> Map[(x-#[[1]])^2+(y-#[[2]])^2&,hull]]],{x,#[[1]]},{y,#[[2]]},
>> >> Gradient->{"FiniteDifference","DifferenceOrder"->2}]&[
>> >> Mean[hull]]]]
>> >>
>> >> Out[69]=
>> >> {0.04 Second,{0.66286,{x->0.486905,y->0.461467}}}
>> >>
>> >>
>> >> My thanks to Bobby Treat, Rob Knapp, and Tom Burton for bringing to my
>> >> attention some of the finer points regarding speed and avoidance of
>> >> premature convergence by FindMinimum.
>> >>
>> >>
>> >> Daniel Lichtblau
>> >> Wolfram Research
>> >
>> >
>> > Another method would be to construct the gradient explicitly; I used a
>> > simple heuristic formula
>> >
>> > (D[Max[f1,...,fn], x] /. x->x0) == (D[f1, x] /. x->x0) /;
>> > f1 > f2,...,f1 > fn,
>> > (D[Max[f1,...,fn], x] /. x->x0) == (D[(f1+f2)/2, x] /. x->x0) /;
>> > f1 == f2, f2 > f3,...,f2 > fn.
>> >
>> > Here's what I get for the same set of points:
>> >
>> > In[1]:=
>> > SeedRandom[1];
>> > points = Partition[Table[Random[], {1000}], 2];
>> >
>> > goal[x_, y_] = Max[Norm[{x, y} - #]& /@ points];
>> >
>> > grad = Compile[
>> > {x, y, {points, _Real, 2}},
>> > Module[
>> > {n = Length@points, Ld = Norm[{x, y} - #]& /@ points,
>> > Ldd, i1, i2},
>> > {i2, i1} = Ordering[Ld, -2];
>> > Ldd = ({x, y} - points[[#]])/Ld[[#]]& /@ {i1, i2};
>> > If[Chop[Ld[[i1]] - Ld[[i2]]] != 0,
>> > Ldd[[1]],
>> > (Ldd[[1]] + Ldd[[2]])/2
>> > ]],
>> > {{_Ordering, _Integer, 1}}
>> > ];
>> >
>> > FindMinimum[goal[x, y],
>> > {x, Mean[points][[1]]}, {y, Mean[points][[2]]},
>> > Method -> Gradient, Gradient :> grad[x, y, points]]
>> >
>> > Out[5]=
>> > {0.662817, {x -> 0.486845, y -> 0.461469}}
>> >
>> > This seems to work quite well for any starting point, but might fail
>> > if there are triple points where f1[x0]==f2[x0]==f3[x0].
>> >
>> > The effect of DifferenceOrder is probably version specific; I tried
>> > all combinations with different methods like
>> > FindMinimum[...,Method->Gradient,Gradient->{FiniteDifference,DifferenceOrder->4}]
>> > but changing DifferenceOrder didn't seem to have any effect.
>> >
>> > Maxim Rytin
>> > m.r at inbox.ru
>> >
>> >
>> >
>
>
>
--
DrBob at bigfoot.com
www.eclecticdreams.net
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