Quartic Riemannian Adaptive Regularization With Cubics for Radar Waveform Design

The ambiguity function is a useful tool for assessing the response performance of radar waveforms. This article addresses the problem of shaping the radar ambiguity function by designing the unimodular transmit waveform. We model the waveform design process as a nonconvex quartic optimization problem aimed at minimizing the disturbance power of scatters in specific range-Doppler bins. Rather than using conventional methods such as approximating and relaxing the objective function and constraints, we address the resultant optimization problem under the Riemannian manifold optimization framework and introduce a novel gradient-based algorithm called quartic Riemannian adaptive regularization with cubics. The developed algorithm applies the Riemannian gradient and Riemannian Hessian of the objective function to conduct iterative operations that gradually decrease the cost function value. The proposed algorithm has a lower iteration complexity bound in comparison to the classical Riemannian trust region method based on the second-order gradient. Numerical experiments demonstrate that our developed method can achieve a relatively higher signal-to-interference ratio with a faster convergence speed.