Weighted Geometric Mean, Minimum Mediated Set, and Optimal Simple Second-Order Cone Representation

TL;DR

This paper investigates optimal simple second-order cone representations for weighted geometric means, establishing bounds, algorithms, and applications in optimization and quantum information.

Contribution

It introduces bounds and algorithms for simple second-order cone representations of weighted geometric means, including exact solutions for bivariate cases and heuristics for general cases.

Findings

Exact size of optimal representations for bivariate means

Algorithms for computing approximate representations

Applications in polynomial, matrix optimization, and quantum info

Abstract

We study optimal simple second-order cone representations (a particular subclass of second-order cone representations) for weighted geometric means, which turns out to be closely related to minimum mediated sets. Several lower and upper bounds on the size of optimal simple second-order cone representations are proved. In the case of bivariate weighted geometric means (equivalently, one dimensional mediated sets), we are able to prove the exact size of an optimal simple second-order cone representation and give an algorithm to compute one. In the genenal case, fast heuristic algorithms and traversal algorithms are proposed to compute an approximately optimal simple second-order cone representation. Finally, applications to polynomial optimization, matrix optimization, and quantum information are provided.