Approximation of the weighted maximin dispersion problem over Lp-ball: SDP relaxation is misleading

TL;DR

This paper analyzes the weighted maximin dispersion problem over Lp-balls, revealing that SDP relaxation can be misleading and proposing a simpler, more effective approach with better practical performance.

Contribution

The paper demonstrates the limitations of SDP relaxation for this problem and introduces a straightforward alternative that improves both theoretical bounds and practical results.

Findings

SDP relaxation provides a limited approximation bound.

Replacing SDP solution with a scalar matrix enhances performance.

The new approach outperforms the SDP-based method in practice.

Abstract

Consider the problem of finding a point in a unit $n$-dimensional $\ell_p$-ball ($p\ge 2$) such that the minimum of the weighted Euclidean distance from given $m$ points is maximized. We show in this paper that the recent SDP-relaxation-based approximation algorithm [SIAM J. Optim. 23(4), 2264-2294, 2013] will not only provide the first theoretical approximation bound of $\frac{1-O\left(\sqrt{ \ln(m)/n}\right)}{2}$, but also perform much better in practice, if the SDP relaxation is removed and the optimal solution of the SDP relaxation is replaced by a simple scalar matrix.