Unconstrained Optimization of Real Functions in Complex Variables

Nonlinear optimization problems in complex variables are frequently encountered in applied mathematics and engineering applications such as control theory, signal processing and electrical engineering. Optimization of these problems often requires a firstor second-order approximation of the objective function to generate a new step or descent direction. However, such methods cannot be applied to real functions in complex variables because they are necessarily nonanalytic in their argument, i.e., the Taylor series expansion in their argument alone does not exist. To overcome this problem, the objective function is often redefined as a function of the real and imaginary parts of its complex argument so that standard optimization methods can be applied. We show that real functions in complex variables do have a Taylor series expansion in complex variables, which we then use to generalize existing optimization methods for both general nonlinear optimization problems and nonlinear least squares problems. We then apply these methods to a number of case studies which show that complex Taylor expansions can lead to greater insight in the structure of the problem and that this structure can often be exploited to improve computational complexity and storage cost.