Quadratically constrained quadratic programming (QCQP) appears widely in engineering applications such as wireless communications and networking and multiuser detection with examples like the MAXCUT problem and boolean optimization. A general QCQP problem is NP-hard. We propose a penalty formulation for the QCQP problem based on semidefinite relaxation. Under suitable assumptions we show that the optimal solutions of the penalty problem are the same as those of the original QCQP problem if the penalty parameter is sufficiently large. Then, to solve the penalty problem, we present a proximal point algorithm and an update rule for the penalty parameter. Numerically, we test our algorithm on two well-studied QCQP problems. The results show that our proposed algorithm is very effective in finding high-quality solutions.

中文翻译：

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二次约束二次规划的正半定罚方法

二次约束二次编程（QCQP）在工程应用中广泛出现，例如无线通信和网络以及多用户检测等工程应用，例如MAXCUT问题和布尔优化。一般的QCQP问题是NP难题。我们提出基于半定松弛的QCQP问题的惩罚公式。在适当的假设下，我们表明如果惩罚参数足够大，惩罚问题的最优解与原始QCQP问题的最优解相同。然后，为了解决惩罚问题，我们提出了惩罚算法的近点算法和更新规则。在数值上，我们在两个经过充分研究的QCQP问题上测试了我们的算法。结果表明，本文提出的算法在寻找高质量解中非常有效。