一种单因子分解预测-校正长步弧搜索方法及其曲率放大变体用于具有齐次自对偶嵌入的半定规划
A One-Factorization Predictor-Corrector Long-Step Arc-Search Method and a Curvature-Amplified Variant for Semidefinite Programming with a Homogeneous Self-Dual Embedding
- Institute of Science Tokyo（东京科学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对半定规划，提出单因子分解弧搜索方法及曲率放大变体，避免二次因子分解，在齐次自对偶嵌入下达到O(n log(1/ε))迭代复杂度，并显著降低计算时间。
AI中文摘要：
在预测-校正弧搜索方法中，首先在预测弧上选择一个点，然后在该点计算校正量。由于Karush-Kuhn-Tucker（KKT）矩阵随所选点变化，这种校正可能需要在每次迭代中进行第二次因子分解。我们提出了一种用于半定规划（SDP）的单因子分解弧搜索方法（OFAS），该方法避免了第二次因子分解，在弧步后保留校正量，并且每次迭代仅使用一次KKT因子分解。关键在于弧上所选点处互补性失配的三项分解，这使得校正可以通过使用当前迭代处的因子分解的线性系统解来组装。我们还提出了一种曲率放大变体（CA-OFAS），它放大了二阶弧项。所提出的方法采用齐次自对偶嵌入进行表述。对于宽长步邻域，我们证明了这两种方法在$O(n \log(1/\varepsilon))$次迭代内将齐次互补性和残差降低到$\varepsilon$以下，其中$n$为$n\times n$ SDP矩阵变量。数值实验表明，单因子分解结构和弧步后的校正与计算时间的减少相关。与Mehrotra型方法相比，CA-OFAS在SDPLIB上将几何平均计算时间减少了17%，在神经网络验证SDP问题上减少了13%。
英文摘要：
In a predictor-corrector arc-search method, a point on the predictor arc is first selected and a corrector is then computed at that point. Since the Karush-Kuhn-Tucker (KKT) matrix changes with the selected point, this correction may require a second factorization in each iteration. We propose a one-factorization arc-search method (OFAS) for semidefinite programming (SDP) that avoids this second factorization, retains the corrector after the arc step, and uses only one KKT factorization per iteration. The key is a three-term decomposition of the complementarity mismatch at the selected point on the arc, which allows the correction to be assembled from linear-system solutions using the factorization at the current iterate. We also propose a curvature-amplified variant (CA-OFAS) that amplifies the second-order arc term. The proposed methods are formulated with a homogeneous self-dual embedding. For a wide long-step neighborhood, we prove that both methods reduce the homogeneous complementarity and residuals below $\varepsilon$ in $O(n \log(1/\varepsilon))$ iterations for an $n$-by-$n$ SDP matrix variable. Numerical experiments show that the one-factorization structure and the correction after the arc step are associated with reductions in computation time. Compared with the Mehrotra-type method, CA-OFAS reduces the geometric-mean computation time by 17% on SDPLIB and 13% on neural network verification SDP problems.
