On Efficient Approximation of the Maximum Distance to A Point Over an Intersection of Balls

TL;DR

This work tackles the NP-hard problem of maximizing the distance $\|x-C_0\|$ over the intersection of balls $\mathcal{Q}$. It develops a convex-analytic framework using $h(x)$ and $g_{\lambda}(x)$ to define a family of convex sets $\mathcal{Q}_{R^2}$ whose containment or intersection properties reveal the optimal distance $R_0$; a three-case theorem governs the location of the critical point $\mathcal{H}^*$. The paper further introduces forward and reverse sequences of intersections (including a seed concept) to characterize and approach $R_0$, proving exponential convergence of the radii and establishing a volume-based, randomized method for approximating $R_0$ while noting the NP-hardness of exact volume computation. Numerical results illustrate the theory and demonstrate the practical viability of the proposed approximation approach. Potential extensions include applying the framework to polytope maximization and exploring log-barrier inspired formulations for broader classes of feasible sets.

Abstract

In this paper we study the NP-Hard problem of maximizing the distance over an intersection of balls to a given point. We expand the results found in \cite{funcos1}, where the authors characterize the farthest in an intersection of balls $\mathcal{Q}$ to the given point $C_0$ by constructing some intersection of halfspaces. In this paper, by slightly modifying the technique found in literature, we characterize the farthest in an intersection of balls $\mathcal{Q}$ with another intersection of balls $\mathcal{Q}_1$. As such, going backwards, we are naturally able to find the given intersection of balls $\mathcal{Q}$ as the max indicator intersection of balls of another one $\mathcal{Q}_{-1}$. By repeating the process, we find a sequence of intersection of balls $(\mathcal{Q}_{i})_{i \in \mathbb{Z}}$, which has $\mathcal{Q}$ as an element, namely $\mathcal{Q}_{0}$ and show that $\mathcal{Q}_{-\infty} = \mathcal{B}(C_0,R_0)$ where $R_0$ is the maximum distance from $C_0$ to a point in $\mathcal{Q}$. As a final application of the proposed theory we give a polynomial algorithm for computing the maximum distance under an oracle which returns the volume of an intersection of balls, showing that the later is NP-Hard. Finally, we present a randomized method %of polynomial complexity which allows an approximation of the maximum distance.

Figures (7)

• Figure 1: The given intersection of balls with green, few instances of the family of max indicator ball polyedra with blue, the member of the family corresponding to the maximum distance
• Figure 2: Close-up and enhancement of Figure \ref{['fig1']}. The given intersection of balls with green filling and green boundary, few instances of the family of max indicator ball polyedra with blue boundary and the member of the family corresponding to the maximum distance with cyan filling and magenta boundary. The point $C_0$ is the black cross.
• Figure 3: The given intersection of balls with green, the centers generated by Procedure A with red. These can be obtained without the knowledge of the max distance $R_0$, although we plot the circles with the correct radius.
• Figure 4: Close-up of Figure \ref{['fig3']}. The formed polytope is $\mathcal{Q}_{R_0}^{\infty}$
• Figure 5: The given intersection of balls with green, the limit case with magenta and with blue intersection of balls which evolve under Procedure A into the green intersection of balls. Note the confirmation of Remark \ref{['R6']}

Theorems & Definitions (18)

• remark 1
• remark 2
• remark 3
• remark 4
• remark 5
• remark 6
• theorem 1
• proof
• corollary 1
• proof