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SONC Optimization and Exact Nonnegativity Certificates via Second-Order Cone Programming

Victor Magron, Jie Wang

TL;DR

This work establishes a constructive $SOC$ representation for the $SONC$ cone, enabling efficient optimization via $SOCP$ for unconstrained polynomial optimization and enabling exact nonnegativity certificates through a hybrid numeric-symbolic approach. It introduces ${\mathscr{T}}$-rational mediated sets and a $SOBS$-based decomposition framework to connect $SONC$ with second-order cone lifts, yielding practical algorithms for lower bounds and certificate generation. The proposed pipeline includes a PN-polynomial reformulation, a simplex-cover strategy, a rounding-projection based exact certification, and comprehensive numerical experiments showing scalability to high-dimensional, high-degree problems with competitive accuracy relative to existing methods. The results demonstrate the viability of SOC-based certificates for sparse polynomials and point to extensions to constrained settings and improved bound quality through richer $SONC$-based representations.

Abstract

The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using SOCs, given that they have a strong expressive ability. In this paper, we prove constructively that the cone of sums of nonnegative circuits (SONC) admits a SOC representation. Based on this, we give a new algorithm for unconstrained polynomial optimization via SOC programming. We also provide a hybrid numeric-symbolic scheme which combines the numerical procedure with a rounding-projection algorithm to obtain exact nonnegativity certificates. Numerical experiments demonstrate the efficiency of our algorithm for polynomials with fairly large degree and number of variables.

SONC Optimization and Exact Nonnegativity Certificates via Second-Order Cone Programming

TL;DR

This work establishes a constructive representation for the cone, enabling efficient optimization via for unconstrained polynomial optimization and enabling exact nonnegativity certificates through a hybrid numeric-symbolic approach. It introduces -rational mediated sets and a -based decomposition framework to connect with second-order cone lifts, yielding practical algorithms for lower bounds and certificate generation. The proposed pipeline includes a PN-polynomial reformulation, a simplex-cover strategy, a rounding-projection based exact certification, and comprehensive numerical experiments showing scalability to high-dimensional, high-degree problems with competitive accuracy relative to existing methods. The results demonstrate the viability of SOC-based certificates for sparse polynomials and point to extensions to constrained settings and improved bound quality through richer -based representations.

Abstract

The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using SOCs, given that they have a strong expressive ability. In this paper, we prove constructively that the cone of sums of nonnegative circuits (SONC) admits a SOC representation. Based on this, we give a new algorithm for unconstrained polynomial optimization via SOC programming. We also provide a hybrid numeric-symbolic scheme which combines the numerical procedure with a rounding-projection algorithm to obtain exact nonnegativity certificates. Numerical experiments demonstrate the efficiency of our algorithm for polynomials with fairly large degree and number of variables.

Paper Structure

This paper contains 20 sections, 19 theorems, 36 equations, 6 figures, 7 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. SONC and sums of binomial squares
  4. Rational mediated sets
  5. Decomposing SONC with binomial squares
  6. SOC representations of SONC cones
  7. SONC optimization via SOCP
  8. Conversion to PN-polynomials
  9. Compute a simplex cover
  10. Exact SOBS decompositions of SONC polynomials
  11. The procedure ${\tt ExactSOBS}$
  12. Arithmetic complexity
  13. Numerical experiments
  14. Computing a lower bound via SONC optimization
  15. Standard simplex
  16. ...and 5 more sections

Key Result

Theorem 1

For ${\mathscr{A}}\subseteq(2{\mathbb{N}})^n$, ${\mathscr{B}}_1\subseteq\hbox{\rm{conv}}({\mathscr{A}})\cap(2{\mathbb{N}})^n$ and ${\mathscr{B}}_2\subseteq\hbox{\rm{conv}}({\mathscr{A}})\cap({\mathbb{N}}^n\backslash(2{\mathbb{N}})^n)$ with ${\mathscr{A}}\cap{\mathscr{B}}_1=\varnothing$, the convex c

Figures (6)

  • Figure 1: Relationship of different classes of polynomials
  • Figure 2: Running time for the standard simplex case
  • Figure 3: Relative optimality gap for the standard simplex case
  • Figure 4: Running time for the general simplex case
  • Figure 5: Relative optimality gap for the general simplex case
  • ...and 1 more figures

Theorems & Definitions (31)

  • Theorem 1
  • Definition 2
  • Theorem 3
  • Example 4
  • Conjecture 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Theorem 9
  • Example 10
  • ...and 21 more