AI 中文总结
研究求解右侧数据变化的二阶锥规划的计算复杂性，提出在前一轮解处热启动牛顿法，证明其在特定条件下的收敛迭代次数及总成本，通过实验证实每轮有显著加速。
AI 中文摘要
我们分析了求解一系列相关二阶锥规划（SOCP）的计算复杂性，这些规划的右侧数据 \(b_t\) 在各轮之间变化。标准的原始对偶内点算法从冷启动开始，每轮求解成本为 \(\tilde{O}(n^{2.5}\log(1/\epsilon))\)。我们表明，当每轮扰动 \(\|b_t - b_{t - 1}\|_2\) 由特定问题阈值 \(\delta\) 界定时，在前一轮解 \(x^*_{t - 1}\) 处热启动的牛顿法在 \(O(\log\log(1/\epsilon))\) 次迭代中收敛到 \(x^*_t\) 达到精度 \(\epsilon\)。在 \(T\) 轮上，总成本为 \(\tilde{O}(n^{2.5}\log(1/\epsilon)+T n^2 \log\log(1/\epsilon))\)，而每轮冷启动的成本为 \(\tilde{O}(T n^{2.5}\log(1/\epsilon))\)；对于大 \(T\)，每轮加速比为 \(\Theta(\sqrt{n}\,\log(1/\epsilon)/\log\log(1/\epsilon))\)。该论证结合了中心路径最优值的无穷小局部范数灵敏度界、自和谐有限差分推论以及牛顿法在自和谐障碍上的标准二次收敛域。局部范数公式避免了胖约束矩阵欧几里得灵敏度界的秩亏问题。在 \(n = 50\)，\(p = 100\) 的有界 SOCP 上进行的多种子实验证实了在预测范围内每轮有 30 - 70 倍的加速。
英文摘要
We analyze the computational complexity of solving a sequence of related second-order cone programs (SOCPs) whose right-hand-side data $b_t$ varies between rounds. The standard primal-dual interior-point algorithm solves each round at cost $\tilde{O}(n^{2.5}\log(1/ε))$ from a cold start. We show that when the per-round perturbation $\|b_t - b_{t-1}\|_2$ is bounded by a problem-specific threshold $δ$, Newton's method warm-started at the previous round's solution $x^*_{t-1}$ converges to $x^*_t$ to accuracy $ε$ in $O(\log\log(1/ε))$ iterations. Over $T$ rounds the total cost is $\tilde{O}(n^{2.5}\log(1/ε) + T n^2 \log\log(1/ε))$, compared to $\tilde{O}(T n^{2.5}\log(1/ε))$ for cold start at each round; the per-round speedup for large $T$ is $Θ(\sqrt{n}\,\log(1/ε)/\log\log(1/ε))$. The argument combines an infinitesimal local-norm sensitivity bound on the central-path optimum, a self-concordant finite-difference corollary, and the standard quadratic-convergence basin of Newton's method on a self-concordant barrier. The local-norm formulation circumvents the rank-deficiency issues of Euclidean sensitivity bounds for fat constraint matrices. A multi-seed experiment on bounded SOCPs with $n=50$, $p=100$ confirms a 30-70x per-round speedup across the predicted regime.
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