Convex Quaternion Optimization for Signal Processing: Theory and Applications
This provides a foundational theory for convex quaternion optimization, addressing a gap in signal processing for researchers and practitioners, though it is incremental in extending existing complex and real optimization methods.
The paper tackled the underdeveloped theory of convex quaternion optimization for signal processing by establishing a systematic framework using generalized Hamilton-real calculus, resulting in discriminant theorems and criteria for convex and strongly convex quaternion functions, along with an optimality theorem demonstrated through three applications.
Convex optimization methods have been extensively used in the fields of communications and signal processing. However, the theory of quaternion optimization is currently not as fully developed and systematic as that of complex and real optimization. To this end, we establish an essential theory of convex quaternion optimization for signal processing based on the generalized Hamilton-real (GHR) calculus. This is achieved in a way which conforms with traditional complex and real optimization theory. For rigorous, We present five discriminant theorems for convex quaternion functions, and four discriminant criteria for strongly convex quaternion functions. Furthermore, we provide a fundamental theorem for the optimality of convex quaternion optimization problems, and demonstrate its utility through three applications in quaternion signal processing. These results provide a solid theoretical foundation for convex quaternion optimization and open avenues for further developments in signal processing applications.