Quadratic Optimization for Unimodular Sequence Design via an ADPM Framework

This paper deals with the NP-hard quadratic optimization problem under similarity and constant modulus constraints. A computationally efficient iterative algorithm based on the Alternating Direction Penalty Method (ADPM) framework is proposed for continuous and discrete phase cases. In each iteration, it converts the considered problem into two subproblems with closed-form solutions via an introduced auxiliary variable, while locally increasing the penalty factor involved in the ADPM framework. The proposed algorithm is proven to converge for any initialization under some mild conditions and avoids the non-convergence problem of the Alternating Direction Method of Multipliers (ADMM) when handling the NP-hard problems. It ensures that the obtained solution fulfills the Karush-Kuhn-Tucker (KKT) conditions for the continuous phase case. To further refine the ADPM solution, a joint approach involving both the ADPM and the coordinate descent frameworks is introduced. The extension that solves the quadratic optimization problem incorporating more complicated constraints is also developed. Finally, two radar waveform design examples are presented to demonstrate that the proposed algorithms can outperform their counterparts by providing better objective values with relatively low polynomial computational complexities.