  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.
