发表机构
Facilty of Mathematics and Computer Science, University of Science, Vietnam National University Ho Chi Minh City（越南国立大学胡志明市科大学数学与计算机学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出证书耦合DCA算法，利用Krylov子空间和谱信息解决非凸信任域子问题，实现早期结构逃逸与渐进全局认证，数值实验验证了其高效性。
AI 中文摘要
我们通过一种证书耦合的差凸算法研究非凸信任域子问题，该算法在达到精确一阶平稳性之前融入谱信息。该方法在DCA迭代中维持一个持久的随机Krylov子空间。在接近平稳性时，足够负的移位瑞利商产生一个显式的可行修正，使目标值定量下降；否则，完成指定的Lanczos深度以高概率提供近似的半正定证书。同一Krylov子空间还为后续精度阶段选择DC曲率参数提供了最大特征值的认证上界。我们证明了信任域上的均匀负曲率修正结果，涵盖近正交硬情形，以及固定容差下的有限终止和工作量界限，以及定量的目标间隙证书。在渐进精度调度下，认证的目标值收敛到全局最优，迭代在距离上收敛到全局解集，而Krylov子空间在修正和精度阶段中得以保留。数值实验阐明了分析所确立的现象：负曲率修正可以在相应的收敛后谱修正之前发生，定量下降界在所有测试的近正交实例中得到满足，并且保留Krylov子空间相对于受控比较中重启谱过程，将谱矩阵-向量乘积的中位数减少了约2.6倍。
英文摘要
We study the nonconvex trust-region subproblem by a certificate-coupled difference-of-convex algorithm that incorporates spectral information before exact first-order stationarity is attained. The method maintains a persistent randomized Krylov subspace together with the DCA iterates. Near stationarity, a sufficiently negative shifted Rayleigh quotient yields an explicit feasible correction with a quantitative decrease in the objective value; otherwise, completion of a prescribed Lanczos depth provides, with high probability, an approximate positive-semidefiniteness certificate. The same Krylov subspace also provides a certified upper bound on the largest eigenvalue for selecting the DC curvature parameter at subsequent accuracy stages. We prove a uniform negative-curvature correction result over the trust region, covering the near-orthogonal hard-case regime, together with finite-termination and work bounds at fixed tolerances and a quantitative objective-gap certificate. Under a progressive accuracy schedule, the certified objective values converge to the global optimum and the iterates converge in distance to the global solution set, while the Krylov subspace is retained across corrections and accuracy stages. Numerical experiments illustrate the phenomena established by the analysis: negative-curvature corrections can occur before the corresponding post-convergence spectral corrections, the quantitative decrease bound is satisfied in all tested near-orthogonal instances, and retaining the Krylov subspace reduces the median number of spectral matrix-vector products by approximately a factor of $2.6$ relative to restarting the spectral process in the controlled comparison.
Comments25 pages, 3 figures, submitted
