arXiv:2102.02763v1 [math.OC] 4 Feb 2021 1 A general framework for constrained convex quaternion optimization Julien Flamant, Sebastian Miron, and David Brie 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 origi- nal quaternion optimization problem and its augmented real- domain counterpart. This new framework simultaneously pro- vides 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 implemen- tation of quaternion-domain optimization procedures. Index Terms—quaternion convex optimization; widely affine constraints; optimality conditions; alternating direction method of multipliers; I. I NTRODUCTION T HE USE of quaternion representations is becoming prevalent in many fields, including color imaging [1]–[3], robotics [4], attitude control and estimation [5], [6] polarized signal processing [7]–[9], rolling bearing fault diagnosis [10], computer graphics [11], among others. Compared to conven- tional real and complex models, quaternion algebra permits unique insights into the physics and the geometry of the problem at hand, while preserving a mathematically sound framework. In most applications, quaternions, besides provid- ing elegant and compact algebraic representations, enable a reduction of the number of parameters. For instance, it has been recently shown [12], [13] that quaternion convolutional neural networks (QCNNs) achieve better performance than conventional real CNNs while using fewer parameters. Most problems involving quaternion models can be cast as the minimization of a real-valued function of quaternion vari- ables. Unfortunately, quaternion-domain optimization faces immediatly a major obstacle: quaternion cost functions being real-valued, they are not differentiable according to quaternion analysis [14] – just like real-valued functions of complex variables are not differentiable according to complex analysis [15]. Thus, for a long time, the intrinsic quaternion nature J. Flamant, S. Miron and D. Brie are with the Universit´ e de Lor- raine, CNRS, CRAN, F-54000 Nancy, France. Corresponding author julien.flamant@cnrs.fr. Authors acknowledge funding support of CNRS and GDR ISIS through project OPENING. of quaternion optimization problems has been disregarded by reformulating them as optimization problems over the real field. This procedure, however, has two major drawbacks: (i) it leads to the loss of the quaternion structure that naturally encodes the physics of the problem at hand, and (ii) it generates cumbersome expressions in the real domain. Recently, a crucial step towards quaternion-domain opti- mization has been made with the development of the theory of HR-calculus [16]–[19]. This new framework establishes a complete set of differentiation rules, encouraging the system- atic development of quaternion-domain algorithms. The HR- calculus can be seen as the generalization to quaternions of the CR-calculus [20], which has enabled the formulation of several important complex-domain algorithms [21], [22]. The theory of HR-calculus has led to the development of multiple quaternion-domain algorithms, notably in adaptive filtering [23]–[25], low-rank quaternion matrix and tensor completion [26]–[28] and quaternion neural networks [29], [30]. The increasing number of applications of quaternion algorithms calls for a general methodology dedicated to quaternion op- timization, in order to design, analyze and implement new efficient quaternion-domain algorithms. To this aim, the present paper proposes a general frame- work for solving constrained convex quaternion optimization problems in the quaternion domain. We restrict ourselves to the convex case as it allows to provide strong mathematical guarantees. The proposed approach relies on two key ingre- dients: the generalized HR-calculus to compute derivatives of cost functions defined in terms of quaternion variables, and the systematic use of equivalences between the original quaternion optimization problem and its augmented real-domain counter- part. This allows to provide solid theoretical foundations for our work; it also reveals specificities of quaternion-domain optimization, such as the widely affine equality constraints that naturally arise in constrained quaternion problems. The contributions of this paper 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 La- grangian duality and optimality conditions; (iii) we develop the quaternion alternating direction method of multipliers (Q- ADMM) as a versatile quaternion optimization algorithm. The paper is organized as follows. In Section II we review the necessary material regarding quaternion variables, their dif- ferent representations and discuss the general affine constraint in the quaternion domain. Section III describes generalized HR-calculus and its particular properties in the case of quater- nion cost functions. Section IV introduces the main theoretical tools for quaternion convex optimization problems, including