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.

中文翻译：

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通过 ADPM 框架对单模序列设计进行二次优化

本文处理相似性和恒模约束下的NP-hard二次优化问题。针对连续和离散相位情况，提出了一种基于交替方向惩罚法 (ADPM) 框架的计算高效迭代算法。在每次迭代中，它通过引入的辅助变量将所考虑的问题转换为具有封闭形式解决方案的两个子问题，同时局部增加 ADPM 框架中涉及的惩罚因子。所提出的算法被证明在一些温和的条件下可以收敛于任何初始化，并且在处理 NP-hard 问题时避免了乘法器交替方向法 (ADMM) 的不收敛问题。它确保获得的解决方案满足连续相情况的 Karush-Kuhn-Tucker (KKT) 条件。为了进一步完善 ADPM 解决方案，引入了一种涉及 ADPM 和坐标下降框架的联合方法。还开发了解决包含更复杂约束的二次优化问题的扩展。最后，提出了两个雷达波形设计示例，以证明所提出的算法可以通过以相对较低的多项式计算复杂度提供更好的目标值来超越其对应算法。