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Reducing non-negativity over general semialgebraic sets to non-negativity over simple sets

Olga Kuryatnikova, Juan C. Vera, Luis F. Zuluaga

TL;DR

A universal approach to derive new Ps\"atze for general semialgebraic sets from ones developed for simpler sets from ones developed for simpler sets, such as a box, a simplex, or the non-negative orthant is proposed.

Abstract

A non-negativity certificate (NNC) is a way to write a polynomial so that its non-negativity on a semialgebraic set becomes evident. Positivstellensätze (Psätze) guarantee the existence of NNCs. Both, NNCs and Psätze underlie powerful algorithmic techniques for optimization. This paper proposes a universal approach to derive new Psätze for general semialgebraic sets from ones developed for simpler sets, such as a box, a simplex, or the non-negative orthant. We provide several results illustrating the approach. First, by considering Handelman's Positivstellensatz (Psatz) over a box, we construct non-SOS Schmüdgen-type Psätze over any compact semialgebraic set. That is, a family of Psätze that follow the structure of the fundamental Schmüdgen's Psatz, but where instead of SOS polynomials, any class of polynomials containing the non-negative constants can be used, such as SONC, DSOS/SDSOS, hyperbolic or sums of AM/GM polynomials. Secondly, by considering the simplex as the simple set, we derive a sparse Psatz over general compact sets, which does not require any structural assumptions of the set. Finally, by considering Pólya's Psatz over the non-negative orthant, we derive a new non-SOS Psatz over unbounded sets which satisfy some generic conditions. All these results contribute to the literature regarding the use of non-SOS polynomials and sparse NNCs to derive Psätze over compact and unbounded sets. Throughout the article, we illustrate our results with relevant examples and numerical experiments.

Reducing non-negativity over general semialgebraic sets to non-negativity over simple sets

TL;DR

A universal approach to derive new Ps\"atze for general semialgebraic sets from ones developed for simpler sets from ones developed for simpler sets, such as a box, a simplex, or the non-negative orthant is proposed.

Abstract

A non-negativity certificate (NNC) is a way to write a polynomial so that its non-negativity on a semialgebraic set becomes evident. Positivstellensätze (Psätze) guarantee the existence of NNCs. Both, NNCs and Psätze underlie powerful algorithmic techniques for optimization. This paper proposes a universal approach to derive new Psätze for general semialgebraic sets from ones developed for simpler sets, such as a box, a simplex, or the non-negative orthant. We provide several results illustrating the approach. First, by considering Handelman's Positivstellensatz (Psatz) over a box, we construct non-SOS Schmüdgen-type Psätze over any compact semialgebraic set. That is, a family of Psätze that follow the structure of the fundamental Schmüdgen's Psatz, but where instead of SOS polynomials, any class of polynomials containing the non-negative constants can be used, such as SONC, DSOS/SDSOS, hyperbolic or sums of AM/GM polynomials. Secondly, by considering the simplex as the simple set, we derive a sparse Psatz over general compact sets, which does not require any structural assumptions of the set. Finally, by considering Pólya's Psatz over the non-negative orthant, we derive a new non-SOS Psatz over unbounded sets which satisfy some generic conditions. All these results contribute to the literature regarding the use of non-SOS polynomials and sparse NNCs to derive Psätze over compact and unbounded sets. Throughout the article, we illustrate our results with relevant examples and numerical experiments.

Paper Structure

This paper contains 21 sections, 24 theorems, 69 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Main contribution
  3. Applications
  4. Preliminaries
  5. The compact case
  6. Main result
  7. Non-SOS Schmüdgen-type Psätze
  8. SDSOS Schmüd-gen-type Psätze
  9. Importance of explicit upper and lower bounds in non-SOS Psätze
  10. Computational experiments
  11. A semi-sparse Psatz over non-structured sets
  12. Computational experiments
  13. The unbounded case
  14. Strong positivity
  15. A lifting for the unbounded case
  16. ...and 6 more sections

Key Result

Lemma 1

Let $p \in {\mathbb R}[x]$. For any $x \in {\mathbb R}^n$ we have

Figures (2)

  • Figure 1: Nonnegative polynomial classes. Related optimization model is shown in parentheses.
  • Figure 2: Legend applies to both plots. Comparison of bound ($y$-axis) vs. time (s.) obtained with different hierarchies (of increasing degree) for PO problems Example 1 and 2 in ghaddar2016dynamickuang2017alternative. Straight-line progress indicates out-of-time (1800s) or out-of-memory (16GB).

Theorems & Definitions (57)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1: PenaVZ08
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Remark 1
  • ...and 47 more