AI 中文总结
研究约束二次规划问题，开发非负共轭梯度法，将CG封装在原对偶活动集循环中，能在不精确、无矩阵情况下求解，保持\(O(\sqrt{\kappa})\)速率，比其他方法快，可用于解决合成SPD问题。
AI 中文摘要
共轭梯度法（CG）可求解对称正定（SPD）系统\(Ax = b\)，但不保证\(x \geq 0\)。我们开发了非负共轭梯度法，用于求解约束二次规划\(\min_x \frac{1}{2} x^\top A x - b^\top x\)，约束条件为\(x \geq 0\)，\(Bx = c\)，\(A \succ 0\)（\(Bx = c\)仅在等式增强变体中存在）。该求解器将CG封装在原对偶活动集循环中：每一步在自由变量上求解无约束系统，舍弃变为负的变量，重新纳入具有负约化梯度的任何边界变量，并重启CG。这些切换是线性互补问题\((A, -b)\)的主要枢轴，由于\(A\)是\(P\)矩阵，所以定义良好。终止使用Júdice和Pires的块主枢轴构造，采用最小索引的Bland回退，在有限多个外部步骤中达到唯一全局极小值。我们的贡献将此保证扩展到Krylov内部求解所需的不精确、无矩阵情况：一旦CG的残差与决策余量相关，近似循环与精确循环做出相同的原对偶符号决策，并且直接分解可以不变地替换CG。每个内部求解都是无矩阵的：对于\(A = M^\top M\)，算子\(v \mapsto M^\top(Mv)\)需要与\(M\)进行两次乘积，无需\(O(n^2)\)存储，并保持CG的\(O(\sqrt{\kappa})\)速率，\(\kappa = \kappa(A)\)。压缩谱的正则化或预处理可降低迭代次数；岭/Tikhonov分解\(A \mapsto (1 - \alpha)A + \alpha R^\top R\)也确保所需的\(P\)矩阵性质。同时分析的无循环投影梯度参考方法以较慢的\(O(\kappa)\)速率收敛。在合成SPD问题上，该循环最多在6个外部步骤中达到植入的最优解，并且在密集实例上，通过直接自由集求解比Lawson - Hanson和内点求解器快一个数量级。
英文摘要
The conjugate gradient method (CG) solves a symmetric positive definite (SPD) system $Ax=b$, but does not respect $x\geq0$. We develop non-negative conjugate gradients, a solver for the bound-constrained quadratic $\min_x \tfrac12 x^\top A x - b^\top x$ subject to $x\geq0$, $Bx=c$, $A\succ0$, with $Bx=c$ present only in the equality-augmented variant. The solver wraps CG in a primal-dual active-set loop: each step solves the unconstrained system over the free variables, drops variables that turn negative, re-admits any bound variable with negative reduced gradient, and restarts CG. These toggles are the principal pivots of the linear complementarity problem $(A,-b)$, well defined since $A$ is a $P$-matrix. Termination uses the block-principal-pivoting construction of Júdice and Pires, with a least-index Bland fallback, reaching the unique global minimiser in finitely many outer steps. Our contribution carries this guarantee to the inexact, matrix-free regime the Krylov inner solve requires: once CG's residual is tied to the decision margin, the approximate loop makes the same primal and dual sign decisions as the exact one, and a direct factorisation may replace CG unchanged. Each inner solve is matrix-free: for $A=M^\top M$, the operator $v\mapsto M^\top(Mv)$ needs two products with $M$, no $O(n^2)$ storage, and retains CG's $O(\sqrtκ)$ rate, $κ=κ(A)$. Regularisation or preconditioning that compresses the spectrum lowers the iteration count; a ridge/Tikhonov split $A\mapsto(1-α)A+αR^\top R$ also secures the needed $P$-matrix property. A loop-free projected-gradient reference method, analysed alongside, converges at the slower $O(κ)$ rate. On synthetic SPD problems the loop reaches the planted optimum in at most 6 outer steps, and with a direct free-set solve outpaces Lawson--Hanson and an interior-point solver by an order of magnitude on dense instances.
