A general framework for constrained convex quaternion optimization

Abstract

This paper introduces a general framework for solving constrained convex quaternion optimization problems in the quaternion domain. To soundly derive these new results, the proposed approach leverages the recently developed generalized HR-calculus together with the equivalence between the original quaternion optimization problem and its augmented realdomain counterpart. This new framework simultaneously provides rigorous theoretical foundations as well as elegant, compact quaternion-domain formulations for optimization problems in quaternion variables. Our contributions are threefold: (i) we introduce the general form for convex constrained optimization problems in quaternion variables, (ii) we extend fundamental notions of convex optimization to the quaternion case, namely Lagrangian duality and optimality conditions, (iii) we develop the quaternion alternating direction method of multipliers (Q-ADMM) as a general purpose quaternion optimization algorithm. The relevance of the proposed methodology is demonstrated by solving two typical examples of constrained convex quaternion optimization problems arising in signal processing. Our results open new avenues in the design, analysis and efficient implementation of quaternion-domain optimization procedures.

Quaternion matrix decomposition and its theoretical implications

“…Let α * be the positive root of ( 19) and construct ω * = α * (ω a * + ωb * i + ωc * j + ωd * k). Then (17) holds with α = α * , and thus…”

“…Moreover, there are some noticeable steps towards optimizing the corresponding quaternion represented problems. Specifically, Qi et al [30,31] conducted a systematic study on quaternion matrix optimization, and Flamant et al [17] proposed a general framework for constrained convex quaternion optimization. In terms of algorithms in the quaternion domain, affine projection algorithms [37] and learning algorithms [22] based on gradient and Hessian have been proposed and analyzed.…”