On Minimization/Maximization of the Generalized Multi-Order Complex Quadratic Form With Constant-Modulus Constraints

In this paper, we study the generalized problem that minimizes or maximizes a multi-order complex quadratic form with constant-modulus constraints on all elements of its optimization variable. Such a mathematical problem is commonly encountered in various applications of signal processing. We term it as the constant-modulus multi-order complex quadratic programming (CMCQP) in this paper. In general, the CMCQP is non-convex and difficult to solve. Its objective function typically relates to metrics such as signal-to-noise ratio, Cram\'er-Rao bound, integrated sidelobe level, etc., and constraints normally correspond to requirements on similarity to desired aspects, peak-to-average-power ratio, or constant-modulus property in practical scenarios. In order to find efficient solutions to the CMCQP, we first reformulate it into an unconstrained optimization problem with respect to phase values of the studied variable only. Then, we devise a steepest descent/ascent method with fast determinations on its optimal step sizes. Specifically, we convert the step-size searching problem into a polynomial form that leads to closed-form solutions of high accuracy, wherein the third-order Taylor expansion of the search function is conducted. Our major contributions also lie in investigating the effect of the order and specific form of matrices embedded in the CMCQP, for which two representative cases are identified. Examples of related applications associated with the two cases are also provided for completeness. The proposed methods are summarized into algorithms, whose convergence speeds are verified to be fast by comprehensive simulations and comparisons to existing methods. The accuracy of our proposed fast step-size determination is also evaluated.