SPLD polynomial optimization and bounded degree SOS hierarchies

TL;DR

This paper introduces SPLD polynomials and a bounded degree SOS hierarchy to efficiently solve polynomial optimization problems, demonstrating improved performance and applications in convex regression.

Contribution

It defines SPLD polynomials, proposes the BSOS-SPLD hierarchy, and applies these concepts to convex optimization and statistical regression.

Findings

BSOS-SPLD outperforms standard SOS hierarchy on benchmarks.

An exact SOS relaxation for convex SPLD problems is developed.

Application to convex polynomial regression demonstrates practical utility.

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.