  The optimization of real scalar functions of quaternion variables, such as
the mean square error or array output power, underpins many practical
applications. Solutions often require the calculation of the gradient and
Hessian, however, real functions of quaternion variables are essentially
non-analytic. To address this issue, we propose new definitions of quaternion
gradient and Hessian, based on the novel generalized HR (GHR) calculus, thus
making possible efficient derivation of optimization algorithms directly in the
quaternion field, rather than transforming the problem to the real domain, as
is current practice. In addition, unlike the existing quaternion gradients, the
GHR calculus allows for the product and chain rule, and for a one-to-one
correspondence of the proposed quaternion gradient and Hessian with their real
counterparts. Properties of the quaternion gradient and Hessian relevant to
numerical applications are elaborated, and the results illuminate the
usefulness of the GHR calculus in greatly simplifying the derivation of the
quaternion least mean squares, and in quaternion least square and Newton
algorithm. The proposed gradient and Hessian are also shown to enable the same
generic forms as the corresponding real- and complex-valued algorithms, further
illustrating the advantages in algorithm design and evaluation.
