[R] The smallest enclosing ball problem

[Hans W.Borchers]

I wanted to solve the following geometric optimization problem, sometimes 
called the "enclosing ball problem":

    Given a set P = {p_1, ..., p_n} of n points in R^d, find a point p_0 such 
    that max ||p_i - p_0|| is minimized.

A known algorithm to solve this as a Qudratic Programming task is as follows:

    Define matrix C = (p_1, ..., p_n), i.e. coordinates of points in columns,
    and minimize
                    x' C' C x - \sum (p_i' p_i) x_i
    subject to
                    \sum x_1 = 1, x_i >= 0 .

Let x0 = (x0_1, ..., x0_n) be an optimal solution, then the linear combination

    p0 = \sum x0_i p_i

is the center of the smallest enclosing ball, and the negative value of the 
objective function at x0 is the squared radius of the ball.

As an example, find the smallest ball enclosing the points (1,0,0), (0,1,0), 
and (0.5, 0.5, 0.1) in 3-dim. space. We would expect the center of the ball to 
be at (0.5, 0.5, 0), and with radius 1/sqrt(2).

    C <- matrix( c(1, 0, 0.5,
                   0, 1, 0.5,
                   0, 0, 0.1), ncol = 3, byrow = TRUE)

    # We need to define D = 2    C' C, d = (p_i' p_i), A = c(1,1,1), and b = 1
    D <- 2 * t(C) %*% C
    d <- apply(C^2, 2, sum)
    A <- matrix(1, nrow = 1, ncol = 3)
    b <- 1

    # Now solve with solve.QP() in package quadprog ...
    require(quadprog)
    sol1 <- solve.QP(D, d, t(A), b, meq = 1)

    # ... and check the result
    sum(sol1$solution)                  # 1
    p0 <- c(C %*% sol1$solution)        # 0.50  0.50 -2.45
    r0 <- sqrt(-sol1$value)             # 2.55

    # distance of all points to the center
    sqrt(colSums((C - p0)^2))           # 2.55 2.55 2.55

As a result we get a ball such that all points lie on the boundary.
The same happens for 100 points in 100-dim. space (to keep D positive 
definite, n = d is required).
That is a very nice, even astounding result, but not what I had hoped for!

Compare this with another quadratic programming solver in R, for example 
LowRankQP() in the package of the same name.

    require(LowRankQP)
    sol2 <- LowRankQP(D, -d, A, b, u = rep(3, 3), method="LU")

    p2 <- c(C %*% sol2$alpha)   # 5.000000e-01 5.000000e-01 1.032019e-12
    sqrt(colSums((C - p2)^2))   # 0.7071068 0.7071068 0.1000000

The correct result, and we get correct solutions in higher dimensions, too. 
LowRankQP works also if we apply it with n > d, e.g., 100 points in 2- 
or 3-dimensional space, when matrix D is not positive definite -- solve.QP( ) 
does not work in these cases.

*** What do I do wrong in calling solve.QP()? ***

My apologies for this overly long request. I am sending it through the weekend
hoping that someone may have a bit of time to look at it more closely.

Hans Werner