A Neurodynamic Approach to Nonsmooth Quaternion Distributed Convex Optimization With Inequality and Affine Equality Constraints

Quaternions have appeared in many practical fields, such as image processing and data mining, and so on. This paper focuses on designing an efficient quaternion-valued neurodynamic approach (QNA) based on multi-agent systems to solve nonsmooth convex quaternion distributed optimization problems (QDOPs) with inequality and affine equality constraints. Each agent in the system cooperatively solves the minimum of the global objective function through the information of itself and its neighbors. At first, the related convex analysis of the quaternion field is given, which provides a theoretical basis for solving the nonsmooth QDOP. Then the considered QDOP is equivalently transformed by using the connectivity of the communication topology. After that, a distributed QNA is presented, where the adaptive controller is introduced to ensure that the penalty terms with respect to the inequality constraints can self-adjust according to the local states. It is shown that all agents reach a consensus while obtaining the optimal solution to the related QDOP. Finally, a numerical example and an application in dictionary sparse representation of color images based on quaternion are realized to intuitively describe the effectiveness and practical significance of the proposed QNA.