Quaternionic Sums of Squares, Moments, and Quaternionic Polynomial Optimization

Abstract

We develop a moment-quaternionic-sum-of-squares (Moment--QSOS) framework for global polynomial optimization over the standard Hamilton quaternions. The main difficulty is the hybrid algebraic structure of scalar quaternionic variables: multiplication is noncommutative, while scalar evaluations satisfy additional polynomial identities inherited from the four commuting real coordinates of each quaternion. To capture this structure, we introduce quaternionic sums of squares together with real and quaternionic scalar-identity ideals. We establish Archimedean Positivstellens\"atze for real-coefficient quaternionic polynomials and for objectives with quaternionic middle coefficients and real-coefficient constraints; a counterexample shows that the latter constraint hypothesis cannot, in general, be removed for the proposed pure-word certificate cone. On the dual side, we solve the corresponding full real-valued and quaternion-valued moment problems through Riesz--Haviland-type characterizations, and we prove a flat-extension theorem for truncated quaternion-valued moments under an explicit scalar-consistency condition. These results yield convergent Moment--QSOS hierarchies and finite-exactness criteria based on flatness. For quaternionic quadratically constrained quadratic programs (QCQPs), we further prove that the first-order quaternionic moment relaxation has the same optimal value as the first-order relaxation of the equivalent real QCQP. Numerical experiments on quadratic, quartic, and sparse problems, together with applications to quaternion-based maximum margin feature extraction and rotation synchronization, show that the quaternionic formulation attains the same relaxation values as the tested real SOS formulations while requiring substantially shorter computation times in the reported instances.