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.

翻译：凸优化方法在通信与信号处理领域已得到广泛应用，然而四元数优化的理论目前尚未达到复优化和实优化那般完整与系统。为此，基于广义汉密尔顿-实（GHR）微积分，我们建立了面向信号处理的凸四元数优化基础理论，其构建方式与传统复优化和实优化理论保持一致。为严谨起见，我们提出了凸四元数函数的五个判别定理，以及强凸四元数函数的四个判别准则。此外，我们给出了凸四元数优化问题最优性的基本定理，并通过四元数信号处理中的三个应用示例展示了其效用。这些结果为凸四元数优化奠定了坚实的理论基础，并为信号处理应用的进一步发展开辟了道路。