Tight Bounds for the Maximum Distance Over a Polytope to a Given Point

TL;DR

The paper addresses maximizing the distance from a fixed point $C_0$ to a polytope $\mathcal{P}$, a problem that is NP-hard in general. It proposes a constructive relaxation by replacing $\mathcal{P}$ with an intersection $\mathcal{Q}$ of congruent balls inside a circumscribing ball, carefully chosen to preserve the maximizers on the boundary. Iteratively applying this ball-intersection transformation yields a polytope $\mathcal{Q}_{R_0^2}^k$ on which the distance maximization can be solved in polynomial time, providing a nontrivial upper bound for the original problem. Numerical experiments in 2D and up to dimension $n=100$ on the unit hypercube illustrate the method's practicality and its ability to produce bounds; the work also outlines a potential refinement via a sequence of decreasing-radius balls to tighten the bound further.

Abstract

In this paper we study the problem of maximizing the distance to a given point $C_0$ over a polytope $\mathcal{P}$. Assuming that the polytope is circumscribed by a known ball we construct an intersection of balls which preserves the vertices of the polytope on the boundary of this ball, and show that the intersection of balls approximates the polytope arbitrarily well. Then, we use some known results regarding the maximization of distances to a given point over an intersection of balls to create a new polytope which preserves the maximizers to the original problem. Next, a new intersection of balls is obtained in a similar fashion, and as such, after a finite number of iterations, we conjecture, we end up with an intersection of balls over which we can maximize the distance to the given point. The obtained distance is shown to be a non trivial upper bound to the original distance. Tests are made with maximizing the distance to a random point over the unit hypercube up to dimension $n = 100$. Several detailed 2-d examples are also shown.

Figures (5)

• Figure 1: The intersection of balls with red, the max indicator polytopes with blue: three instances of the family, $C_0$ is the green star and the identified maximum distance is given by the black circle. The last vertex to enter the intersection of balls is the farthest from $C_0$
• Figure 2: An example: The initial convex hull with green. The translated convex hull with red. One of the facets approaches the point $C_0$. The adjusted convex hull with blue. Still has one facet closer to $C_0$ than in initial convex hull.
• Figure 3: With green one sees the last polytope and its associated intersection of balls. With green star the point $C_0$ and the black circle show the identified maximum distance.
• Figure 4: A perturbed cube example. Note that the centers of the balls (with green stars) forming the intersection of balls $\mathcal{Q}_{R_0^2}^k$ associated to $\mathcal{P}_{R_0^2}^k$ (with blue) do not contain $C_0$ in their convex hull.
• Figure 5: The obtained error for each entry in the obtained solution. The dimension here is $n = 100$. Our implementation uses the ellipsoid algorithm pedagogically implemented locally for some sub-procedures. Improvements can be made.

Theorems & Definitions (9)

• remark 1
• remark 2
• lemma 1
• proof
• corollary 1
• proof
• theorem 1
• proof
• remark 3