The problem of minimizing the sum of a convex quadratic function and the ratio of two quadratic functions can be reformulated as a Celis–Dennis–Tapia (CDT) problem and, thus, according to some recent results, can be polynomially solved. However, the degree of the known polynomial approaches for these problems is fairly large and that justifies the search for efficient local search procedures. In this paper the CDT reformulation of the problem is exploited to define a local search algorithm. On the theoretical side, its convergence to a stationary point is proved. On the practical side it is shown, through different numerical experiments, that the main cost of the algorithm is a single Schur decomposition to be performed during the initialization phase. The theoretical and practical results for this algorithm are further strengthened in a special case.

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高效的局部搜索程序，解决二次分数规划问题

可以将凸二次函数之和与两个二次函数之比最小化的问题重新表述为Celis-Dennis-Tapia（CDT）问题，因此，根据最近的一些结果，可以用多项式求解。然而，针对这些问题的已知多项式方法的程度相当大，这证明了寻找有效的局部搜索程序的合理性。本文利用CDT对问题的重新表述来定义本地搜索算法。从理论上讲，证明了其收敛到固定点。在实践方面，通过不同的数值实验表明，该算法的主要成本是在初始化阶段要执行的单个Schur分解。