is.inCH returns TRUE if and only if p is contained in
the convex hull of the points given as the rows of M. If p is
a matrix, each row is tested individually, and TRUE is returned if
all rows are in the convex hull.
is.inCH(p, M, verbose = FALSE, ...)A \(d\)-dimensional vector or a matrix with \(d\) columns
An \(r\) by \(d\) matrix. Each row of M is a
\(d\)-dimensional vector.
A logical vector indicating whether to print progress
arguments passed directly to linear program solver
Logical, telling whether p is (or all rows of p are)
in the closed convex hull of the points in M.
The \(d\)-vector p is in the convex hull of the \(d\)-vectors
forming the rows of M if and only if there exists no separating
hyperplane between p and the rows of M. This condition may be
reworded as follows:
Letting \(q=(1 p')'\) and \(L = (1 M)\), if the maximum value of
\(z'q\) for all \(z\) such that \(z'L \le 0\) equals zero (the maximum
must be at least zero since z=0 gives zero), then there is no separating
hyperplane and so p is contained in the convex hull of the rows of
M. So the question of interest becomes a constrained optimization
problem.
Solving this problem relies on the package lpSolve to solve a linear
program. We may put the program in "standard form" by writing \(z=a-b\),
where \(a\) and \(b\) are nonnegative vectors. If we write \(x=(a'
b')'\), we obtain the linear program given by:
Minimize \((-q' q')x\) subject to \(x'(L -L) \le 0\) and \(x \ge 0\). One additional constraint arises because whenever any strictly negative value of \((-q' q')x\) may be achieved, doubling \(x\) arbitrarily many times makes this value arbitrarily large in the negative direction, so no minimizer exists. Therefore, we add the constraint \((q' -q')x \le 1\).
This function is used in the "stepping" algorithm of Hummel et al (2012).
Hummel, R. M., Hunter, D. R., and Handcock, M. S. (2012), Improving Simulation-Based Algorithms for Fitting ERGMs, Journal of Computational and Graphical Statistics, 21: 920-939.