A Moment-QSOS Hierarchy for a Class of Quaternion Polynomial Optimization Problems

TL;DR

This paper develops a hierarchy of semidefinite relaxations for quaternion polynomial optimization, improving scalability and tightness, with demonstrated efficiency on practical problems.

Contribution

It introduces a novel quaternion-specific SOS hierarchy with sparsity and basis strengthening techniques for better scalability and solution quality.

Findings

Provides bounds comparable to existing methods

Reduces computation time and memory usage

Effective on quaternion-based maximum margin and orientation synchronization problems

Abstract

This paper introduces a Moment-Quaternion-Sum-of-Squares (QSOS) hierarchy for solving a class of quaternion polynomial optimization problems. This hierarchy is formulated directly in the quaternion domain and consists of a sequence of semidefinite programming (SDP) relaxations that provide monotonic lower bounds on the optimal value. To improve scalability, we incorporate correlative sparsity, which can significantly reduce the size of the resulting SDPs for large-scale sparse problems. Furthermore, we introduce a strengthened QSOS relaxation, which enhances the tightness of the standard relaxation by enlarging the monomial basis in a controlled manner. Our various Numerical experiments show that our approach provides comparable bounds to existing approaches, while significantly reducing computation time and memory usage. In particular, applications to the quaternion-based maximum margin criterion problem and the classical orientation synchronization problem illustrate the practical effectiveness of the framework.