发表机构
Jebel Quant Research; Faculty of Mathematics, TU Chemnitz（杰贝尔量化研究; 德累斯顿工业大学化学nitz分校数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文重新审视Goldfarb-Idnani方法，分析其在边界约束外的失效问题，通过实验验证复制$C$列会大幅提升失效概率，为该方法的应用提供了关键限制。
AI 中文摘要
Goldfarb和Idnani提出的对偶活动集方法通过每次添加和删除一个约束来求解严格凸二次规划，无需第一阶段。原始-对偶活动集方法和块主元枢轴法则完全跳过迭代过程：它们一次性猜测整个活动集，并通过修复返回符号来调整。对于边界约束，这种猜测是安全的：在候选集上求解的系统是$G^{-1}$的主子矩阵，无论猜测如何都是正定的，该P矩阵性质保证了有限终止。但对于$C^\top x \ge b$，这一假设不成立，这正是我们的核心研究重点。此时，工作集系统为$C_{\mathcal{A}}^\top G^{-1} C_{\mathcal{A}}$，仅当$C_{\mathcal{A}}$满列秩时正定，而这是猜测的属性而非数据的属性。P矩阵性质丧失，结构障碍使有限终止保证失效。严格凸性使该方法仍可使用：KKT条件使候选集得到验证而非信任。在四个约束族的600多个随机实例中，这种失效从未发生；但复制$C$的列会使失效概率升至93%，验证比例从100%降至0%。在真实的仅做多资产组合数据上，活动集达到了494+1个约束中的442个。
英文摘要
The dual active-set method of Goldfarb and Idnani solves the strictly convex quadratic program by adding and dropping one constraint at a time, requiring no phase one. Primal--dual active set and block principal pivoting skip the walk altogether: they guess an entire active set at once and repair it from returning signs. For bound constraints the guess is safe. The system solved on a candidate set is a principal submatrix of $G^{-1}$, positive definite regardless of the guess; this $P$-matrix property guarantees finite termination. For $C^\top x \ge b$ that hypothesis fails, which is our central focus. The working-set system is $C_{\mathcal{A}}^\top G^{-1} C_{\mathcal{A}}$, positive definite only when $C_{\mathcal{A}}$ has full column rank, a property of the guess, not the data. The $P$-matrix property is lost and the structural obstruction invalidates the guarantee. Strict convexity keeps the method usable: KKT conditions make candidate sets certified rather than trusted. Over 600 random instances across four constraint families this failure never occurs. However, duplicating columns of $C$ raises failure rates to 93\%, dropping certified fraction from 100\% to zero. On real long-only portfolio data, the active set reaches 442 of $494 + 1$ constraints.
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