SPLD polynomial optimization and bounded degree SOS hierarchies

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.