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SPLD polynomial optimization and bounded degree SOS hierarchies

Liguo Jiao, Jae Hyoung Lee, Nguyen Bui Nguyen Thao

TL;DR

The paper defines separable plus lower-degree (SPLD) polynomials and develops the BSOS-SPLD hierarchy, a bounded-degree SOS framework that exploits SPLD structure to solve global polynomial optimization efficiently. It proves convergence to the global optimum under mild assumptions and shows exact SOS relaxations for SOS-convex SPLD problems, with practical recovery of solutions. Through numerical experiments on nonconvex SPLD test problems and portfolio optimization, the approach demonstrates superior scalability and accuracy compared to the standard BSOS hierarchy, and it also applies to SOS-convex SPLD polynomial regression. The work advances scalable global optimization for structured high-degree polynomials and opens avenues for large-scale SPLD applications and regression tasks in statistics.

Abstract

In this paper, we introduce a new class of structured polynomials, called separable plus lower degree (SPLD) polynomials. The formal definition of an SPLD polynomial, which extends the concept of SPQ polynomials (Ahmadi et al. in Math Oper Res 48:1316--1343, 2023), is provided. A type of bounded degree SOS hierarchy, referred to as BSOS-SPLD, is proposed to efficiently solve optimization problems involving SPLD polynomials. Numerical experiments on several benchmark problems indicate that the proposed method yields better performance than the standard bounded degree SOS hierarchy (Lasserre et al. in EURO J Comput Optim 5:87--117, 2017). An exact SOS relaxation for a class of convex SPLD polynomial optimization problems is proposed. Finally, we present an application of SPLD polynomials to convex polynomial regression problems arising in statistics.

SPLD polynomial optimization and bounded degree SOS hierarchies

TL;DR

The paper defines separable plus lower-degree (SPLD) polynomials and develops the BSOS-SPLD hierarchy, a bounded-degree SOS framework that exploits SPLD structure to solve global polynomial optimization efficiently. It proves convergence to the global optimum under mild assumptions and shows exact SOS relaxations for SOS-convex SPLD problems, with practical recovery of solutions. Through numerical experiments on nonconvex SPLD test problems and portfolio optimization, the approach demonstrates superior scalability and accuracy compared to the standard BSOS hierarchy, and it also applies to SOS-convex SPLD polynomial regression. The work advances scalable global optimization for structured high-degree polynomials and opens avenues for large-scale SPLD applications and regression tasks in statistics.

Abstract

In this paper, we introduce a new class of structured polynomials, called separable plus lower degree (SPLD) polynomials. The formal definition of an SPLD polynomial, which extends the concept of SPQ polynomials (Ahmadi et al. in Math Oper Res 48:1316--1343, 2023), is provided. A type of bounded degree SOS hierarchy, referred to as BSOS-SPLD, is proposed to efficiently solve optimization problems involving SPLD polynomials. Numerical experiments on several benchmark problems indicate that the proposed method yields better performance than the standard bounded degree SOS hierarchy (Lasserre et al. in EURO J Comput Optim 5:87--117, 2017). An exact SOS relaxation for a class of convex SPLD polynomial optimization problems is proposed. Finally, we present an application of SPLD polynomials to convex polynomial regression problems arising in statistics.

Paper Structure

This paper contains 12 sections, 10 theorems, 86 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. SPLD polynomial optimization
  4. BSOS-SPLD: Bounded degree SOS hierarchies for SPLD polynomial optimization
  5. Numerical experiments
  6. BSOS versus BSOS-SPLD
  7. An application to portfolio optimization problems
  8. A class of convex SPLD polynomial optimization problems
  9. Convex polynomials and SPLD polynomials
  10. Exact SOS relaxations for SOS-convex SPLD polynomial optimization
  11. SOS-convex SPLD polynomial regression
  12. Conclusions and further remarks

Key Result

Proposition 2.1

Lasserre2010Lasserre2015book A polynomial $f\in{\mathbb R}[\textrm{\bf{x}}]_{2d}$ has an SOS decomposition if and only if there exists $Q\in \mathcal{S}^{s(n,d)}_+$ such that

Figures (1)

  • Figure 1:

Theorems & Definitions (24)

  • Proposition 2.1
  • Definition 2.1: SOS-convexity Helton2010Ahmadi2013
  • Lemma 3.1: Krivine--Stengle's positivstellensatz Lasserre2010Krivine1964Stengle1974
  • Theorem 3.1
  • proof
  • Remark 3.1: Comparison with the standard BSOS hierarchy Lasserre2017
  • Theorem 3.2: cf. Lasserre2017
  • proof
  • Lemma 5.1
  • Lemma 5.2
  • ...and 14 more