[R] The smallest enclosing ball problem
Berend Hasselman
bhh at xs4all.nl
Sun Nov 17 12:05:16 CET 2013
On 17-11-2013, at 11:32, Hans W.Borchers <hwborchers at googlemail.com> wrote:
> Berend Hasselman <bhh <at> xs4all.nl> writes:
>> Forgot to forward my answer to R-help.
>>
>> Berend
>
> Thanks, Berend, for thinking about it. \sum xi = 1 is a necessary condition
> to generate a valid geometric solution. The three points in the example are
> very regular and your apporach works, but imagine some random points:
>
> set.seed(8237)
> C <- matrix(runif(9), 3, 3)
> D <- 2 * t(C) %*% C
> d <- apply(C^2, 2, sum)
> A <- diag(3)
> b <- rep(0,3)
>
> require(quadprog)
> sol1 <- solve.QP(D, d, A, b, meq = 0)
> sol1 # now \sum xi = 1is not fulfilled
>
> p0 <- c(C %*% sol1$solution) # 0.3707410 0.3305265 0.2352084
> r0 <- sqrt(-sol1$value) # 0.5495631
>
> # distance of all points to the center
> sqrt(colSums((C - p0)^2)) # 0.5495631 0.3119314 0.5495631
>
> Unfortunately, this is not the smallest enclosing ball.
> LowRankQP will find the true solution with radius 0.3736386 !
>
> require(LowRankQP)
> A <- matrix(1, nrow = 1, ncol = 3)
> b <- 1
>
> sol2 <- LowRankQP(D, -d, A, b, u = rep(1, 3), method="LU")
>
> p2 <- c(C %*% sol2$alpha) # 0.5783628 0.5372570 0.2017087
> sqrt(colSums((C - p2)^2)) # 0.3736386 0.3736386 0.3736386
>
> But the strangest thing is that with \sum xi =1 solve.QP positions all points
> on the boundary, though (in my opinion) no constraint requests it. So the
> question remains:
> *** What do I do wrong in calling solve.QP()? ***
>
> Hans Werner
See my second reply to your original post.
Modify your code with
A <- matrix(rep(1,3),nrow=4,ncol=3,byrow=TRUE)
A[2:4,] <- diag(3)
b <- c(1,0,0,0)
to include constraints x_i>=0 (which LowRankQP includes automatically!) and run solve.QP as follows
sol2 <- solve.QP(D, d, t(A), b, meq = 1)
sol2
p0 <- c(C %*% sol2$solution)
r0 <- sqrt(-sol2$value)
p0
r0
# distance of all points to the center
sqrt(colSums((C - p0)^2))
and the answers now agree with LowRankQP.
Berend
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