  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 $\mathbb{HR}$-calculus together with the equivalence between the
original quaternion optimization problem and its augmented real-domain
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.
