Unconstrained optimization of real functions in complex variables

Complex numbers are a fundamental tool for applied mathematics and many engineering applications such as control theory, signal processing and electrical engineering. Many nonlinear optimization methods use a first- or second-order approximation of an objective function to generate a new step or descent direction. A problem that arises in applying these methods to real functions of complex variables, is that they are necessarily nonanalytic in their argument, i.e. their Taylor series expansion does not exist. A common workaround is to convert the optimization problem to the real domain 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. We then apply these methods to a number of case studies which show that complex Taylor expansions give greater insight in the structure of the problem and that this structure can often be exploited to improve computational complexity and memory cost.status: publishe