Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jie Wang

TL;DR

Several lower and upper bounds on the size of optimal simple second-order cone representations are proved and several applications to polynomial optimization, matrix optimization, and quantum information are provided.

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.

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

TL;DR

Several lower and upper bounds on the size of optimal simple second-order cone representations are proved and several applications to polynomial optimization, matrix optimization, and quantum information are provided.

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.
Paper Structure (19 sections, 22 theorems, 59 equations, 2 figures, 6 tables, 2 algorithms)

This paper contains 19 sections, 22 theorems, 59 equations, 2 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Optimal simple second-order cone representations and minimum mediated sets
  4. More general second-order cone representations
  5. A binary tree representation of successive minimum mediated sequences
  6. Algorithms
  7. Fast heuristic algorithms
  8. Traversal algorithms
  9. A brute-force algorithm
  10. Numerical experiments
  11. Evaluating different heuristic algorithms
  12. Comparison with optimal simple second-order cone representations
  13. Applications
  14. SONC optimization
  15. Matrix optimization
  16. ...and 4 more sections

Key Result

Theorem 6

Let $s_1,\ldots,s_m\in{\mathbb{N}}^*$ be a tuple of integers with $(s_1,\ldots,s_m)=1$. Then

Figures (2)

  • Figure 1: $\{{\boldsymbol{\beta}}_1,{\boldsymbol{\beta}}_2,{\boldsymbol{\beta}}_3\}$ forms an $\{{\boldsymbol{\alpha}}_1,{\boldsymbol{\alpha}}_2,{\boldsymbol{\alpha}}_3\}$-mediated set.
  • Figure 2: The binary tree representation of a successive minimum $(p,q)$-mediated sequence with $p=57,q=11$.

Theorems & Definitions (54)

  • Definition 1
  • Remark 2
  • Remark 3
  • Example 4
  • Example 5
  • Theorem 6
  • proof
  • Example 7
  • Definition 8
  • Lemma 9
  • ...and 44 more