MINLP formulations for continuous piecewise linear function fitting

We consider a nonconvex mixed-integer nonlinear programming (MINLP) model proposed by Goldberg et al. (Comput Optim Appl 58:523–541, 2014. 10.1007/s10589-014-9647-y) for piecewise linear function fitting. We show that this MINLP model is incomplete and can result in a piecewise linear curve that is not the graph of a function, because it misses a set of necessary constraints. We provide two counterexamples to illustrate this effect, and propose three alternative models that correct this behavior. We investigate the theoretical relationship between these models and evaluate their computational performance.

REGION INTERVISIBILITY IN TERRAINS

A polyhedral terrain is the graph of a continuous piecewise linear function defined over the triangles of a triangulation in the xy-plane. Two points on or above a terrain are visible to each other if the line-of-sight does not intersect the space below the terrain. In this paper, we look at three related visibility problems in terrains. Suppose we are given a terrain T with n triangles and two regions R1 and R2 on T, i.e., two simply connected subsets of at most m triangles. First, we present an algorithm that determines, for any constant ∊ > 0, within O(n1+∊m) time and storage whether or not R1 and R2 are completely intervisible. We also give an O(m3n4) time algorithm to determine whether every point in R1 sees at least one point in R2. Finally, we present an O(m2n2 log n) time algorithm to determine whether there exists a pair of points p ∈ R1 and q ∈ R2, such that p and q see each other.