Advances in International Applied Mathematics
Advances in International Applied Mathematics. 2025; 7: (3) ; 10.12208/j.aam.20250020 .
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扬州大学数学学院 江苏扬州
*通讯作者： 陶文清,单位：扬州大学数学学院 江苏扬州；
本文研究了 QR 分解在带等式约束的二次优化问题中的应用，并以 Markowitz 投资组合模型为主要实例。通过对约束矩阵进行正交分解，QR 方法能够在代数上实现约束的消元，将原问题降维为无约束形式，从而避免传统 KKT 系统中不定矩阵的数值不稳定。本文建立了三资产投资组合与四资产发电结构两类算例，分别采用QR方法与KKT方法进行数值对比。结果显示，QR方法将系统条件数降低了一个数量级，显著提升了计算精度与算法鲁棒性。此外，QR分解在几何上提供了清晰的投影解释，增强了约束优化问题的直观性与可解释性。该方法在金融投资、能源结构与其他资源分配类问题中具有推广价值。
This paper investigates the application of the QR decomposition to equality-constrained quadratic optimization problems, with the Markowitz portfolio model serving as the main example. By performing an orthogonal decomposition of the constraint matrix, the QR-based approach eliminates equality constraints algebraically and transforms the original problem into a reduced, unconstrained form, thereby avoiding the numerical instability inherent in the indefinite KKT system. Two representative models are analyzed—a three-asset portfolio and a four-asset power generation structure—accompanied by numerical comparison between two methods. The results demonstrate that the QR method reduces the system condition number by an order of magnitude, thereby significantly enhancing computational accuracy and algorithmic robustness. Besides, QR method provides a clearer geometric interpretation through orthogonal projection, offering an intuitive framework for constrained optimization. The approach is applicable to financial, energy, and other resource-allocation optimization problems.
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