Description

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.

Usage

Arguments

Details

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 ≤ 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) ≤ 0 and x ≥ 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 ≤ 1.

This function is used in the "stepping" algorithm of Hummel et al (2012).

Value

Logical, telling whether p is (or all rows of p are) in the closed convex hull of the points in M.

References

• https://www.cs.mcgill.ca/~fukuda/soft/polyfaq/node22.html

• 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.