The Continuous p-Dispersion Problem in Three Dimensions

TL;DR

This paper introduces an almost-everywhere differentiable optimization model and global optimization algorithm for approximating solutions to the Continuous p-Dispersion Problem with boundary constraints in convex and non-convex polyhedra with respect to any metric in a three-dimensional Euclidean space.

Abstract

The Continuous p-Dispersion Problem (CpDP) with boundary constraints asks for the placement of a fixed number of points in a compact subset of Euclidean space such that the minimum distance between any two points, as well as the points and the boundary of this compact set is maximized. This problem finds applications in facility placement, communication network design, sampling theory, and particle simulation; however, finding optimal solutions is NP-hard and existing algorithms focus on providing approximate solutions in two-dimensional space. In this paper, we introduce an almost-everywhere differentiable optimization model and global optimization algorithm for approximating solutions to the CpDP with boundary constraints in convex and non-convex polyhedra with respect to any metric in a three-dimensional Euclidean space. Our algorithm generalizes two-dimensional dispersion techniques to three dimensions by leveraging orientation, linear-algebraic projections for point-to-face distances, and a ray-casting procedure for point-in-polyhedron testing, enabling optimization in arbitrary convex and non-convex three dimensional polyhedra. We validate the proposed algorithm by comparing with analytical optima where available and empirical benchmarks, observing close agreement with optimal solutions and improvements over empirical benchmarks.

Figures (17)

• Figure 1: All active faces (magenta) of the H-box container for point $c = (1.5, 0.5, 1.5)$ inside the container.
• Figure 2: All active faces (magenta) of the H-box container for point $c = (1.5, 0.5, 2.5)$ outside the container.
• Figure 3: All active footpoints (blue) inside the H-box container with respect to (red) point $(1.5, 0.5, 1.5)$ inside container with activate faces highlighted in pink.
• Figure 4: All active footpoints (blue) inside the H-box container with respect to (red) point $(1.5, 0.5, 2.5)$ outside container with active faces highlighted in purple.
• Figure 5: Overview of the global optimization algorithm. The process demonstrates a two-stage approach: first, Tabu Search is employed to relocate a candidate sphere (red) from an initial, infeasible position to a feasible position (yellow). Second, an Adjust Distance phase maximizes the uniform radius of all spheres to achieve an optimal packing configuration.