We present an algorithm for the minimization of a nonconvex quadratic function subject to linear inequality constraints and a two-sided bound on the 2-norm of its solution. The algorithm minimizes the objective using an active-set method by solving a series of trust-region subproblems (TRS). Underpinning the efficiency of this approach is that the global solution of the TRS has been widely studied in the literature, resulting in remarkably efficient algorithms and software. We extend these results by proving that nonglobal minimizers of the TRS, or a certificate of their absence, can also be calculated efficiently by computing the two rightmost eigenpairs of an eigenproblem. We demonstrate the usefulness and scalability of the algorithm in a series of experiments that often outperform state-of-the-art approaches; these include calculation of high-quality search directions arising in Sequential Quadratic Programming on problems of the CUTEst collection, and Sparse Principal Component Analysis on a large text corpus problem (70 million nonzeros) that can help organize documents in a user interpretable way.

我们提出了一种最小化非凸二次函数的算法，该二次函数受线性不等式约束和其解的2-范数上的两边有界。该算法通过解决一系列信任区域子问题（TRS），使用主动集方法将目标最小化。这种方法之所以有效，是因为TRS的全局解决方案已经在文献中进行了广泛的研究，从而产生了非常有效的算法和软件。我们证明了TRS的非全局极小值或它们的不存在证明，也可以通过计算一个特征问题的两个最右边的特征对来有效地计算，从而扩展了这些结果。我们在一系列通常优于最新方法的实验中证明了该算法的有用性和可扩展性。CUTEst集合，以及针对大型文本语料库问题（7000万个非零）的稀疏主成分分析，可帮助以用户可解释的方式组织文档。