论文标题
关于信托区域子问题的广义兰斯佐斯信任区域的收敛
On convergence of the generalized Lanczos trust-region method for trust-region subproblems
论文作者
论文摘要
广义的兰科斯信托区（GLTR）方法是解决大规模信任区域子问题（TRS）的最流行方法之一。最近，贾和王[Z. Jia和F. Wang，\ emph {Siam J. Optim。，31（2021），第887--914}]考虑了该方法的收敛性，并确定了一些{\ IT在残差，解决方案，解决方案和Largrange乘数上的{\ IT {\ IT}误差。在本文中，我们重新审视GLTR方法的收敛性，并尝试改善这些界限。首先，我们在残差上建立了更清晰的上限。其次，我们在近似和精确解决方案之间的距离上给出了一个新的结合，并证明近似的收敛性与相关的光谱分离无关。第三，我们为Largrange乘数的收敛提供了一些非质子界限，并定义了在Largrange乘数收敛中起重要作用的因素。数值实验证明了我们的理论结果的有效性。
The generalized Lanczos trust-region (GLTR) method is one of the most popular approaches for solving large-scale trust-region subproblem (TRS). Recently, Jia and Wang [Z. Jia and F. Wang, \emph{SIAM J. Optim., 31 (2021), pp. 887--914}] considered the convergence of this method and established some {\it a prior} error bounds on the residual, the solution and the Largrange multiplier. In this paper, we revisit the convergence of the GLTR method and try to improve these bounds. First, we establish a sharper upper bound on the residual. Second, we give a new bound on the distance between the approximation and the exact solution, and show that the convergence of the approximation has nothing to do with the associated spectral separation. Third, we present some non-asymptotic bounds for the convergence of the Largrange multiplier, and define a factor that plays an important role on the convergence of the Largrange multiplier. Numerical experiments demonstrate the effectiveness of our theoretical results.
