Optimization on a Generalized Multi-Order Complex Quadratic Form with Constant-Modulus Constraints

In this paper, we study the problem that minimizes or maximizes a multi-order complex quadratic form with constant-modulus constraints on all elements of the optimization variable. This mathematical problem is commonly encountered in various applications of signal processing, and we term it as the constant-modulus multi-order complex quadratic programming (CMCQP). In general, the CMCQP is non-convex, whose objective typically relates to metrics such as signal-to-noise ratio, Cramer-Ran bound, integrated sidelobe level, etc., and constraints normally correspond to requirements on similarity to desired aspects, peak-to-average power ratio, or constant modulus in practical scenarios. In order to find an efficient solution to the CMCQP, we first reformulate it into an unconstrained optimization problem. Then we devise steepest descent/ascent methods with fast determination on their optimal step sizes. Our contribution also lies in identifying two representative cases for the CMCQP on optimization. The accuracy of our proposed step-size determination is evaluated and the superiority of our proposed algorithms than the existing algorithms is verified.