Convergence of the block Lanczos method for the trust‐region subproblem in the hard case

IF 1.8 3区数学 Q1 MATHEMATICS

Numerical Linear Algebra with Applications Pub Date : 2024-05-07 DOI:10.1002/nla.2561

Bo Feng, Gang Wu

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引用次数: 0

Abstract

SummaryThe trust‐region subproblem (TRS) plays a vital role in numerical optimization, numerical linear algebra, and many other applications. It is known that the TRS may have multiple optimal solutions in the hard case. In [Carmon and Duchi, SIAM Rev., 62 (2020), pp. 395–436], a block Lanczos method was proposed to solve the TRS in the hard case, and the convergence of the optimal objective value was established. However, the convergence of the KKT error as well as that of the approximate solution are still unknown for this method. In this paper, we give a more detailed convergence analysis on the block Lanczos method for the TRS in the hard case. First, we improve the convergence speed of the approximate objective value. Second, we derive the speed of the KKT error tends to zero. Third, we establish the convergence of the approximation solution, and show theoretically that the projected TRS obtained from the block Lanczos method will be close to the hard case more and more as the block Lanczos process proceeds. Numerical experiments illustrate the effectiveness of our theoretical results.

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困难情况下信任区域子问题的分块 Lanczos 方法的收敛性

摘要信任区域子问题（TRS）在数值优化、数值线性代数和许多其他应用中发挥着重要作用。众所周知，TRS 在困难情况下可能有多个最优解。在 [Carmon and Duchi, SIAM Rev., 62 (2020), pp.然而，该方法的 KKT 误差收敛性以及近似解的收敛性仍然未知。本文对块 Lanczos 方法进行了更详细的收敛分析。首先，我们改进了近似目标值的收敛速度。其次，我们得出了 KKT 误差趋于零的速度。第三，我们建立了近似解的收敛性，并从理论上证明了随着分块 Lanczos 过程的进行，由分块 Lanczos 方法得到的投影 TRS 将越来越接近硬情形。数值实验证明了我们理论结果的有效性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Numerical Linear Algebra with Applications 数学-数学

CiteScore

3.40

自引率

2.30%

发文量

审稿时长

12 months

期刊介绍： Manuscripts submitted to Numerical Linear Algebra with Applications should include large-scale broad-interest applications in which challenging computational results are integral to the approach investigated and analysed. Manuscripts that, in the Editor’s view, do not satisfy these conditions will not be accepted for review. Numerical Linear Algebra with Applications receives submissions in areas that address developing, analysing and applying linear algebra algorithms for solving problems arising in multilinear (tensor) algebra, in statistics, such as Markov Chains, as well as in deterministic and stochastic modelling of large-scale networks, algorithm development, performance analysis or related computational aspects. Topics covered include: Standard and Generalized Conjugate Gradients, Multigrid and Other Iterative Methods; Preconditioning Methods; Direct Solution Methods; Numerical Methods for Eigenproblems; Newton-like Methods for Nonlinear Equations; Parallel and Vectorizable Algorithms in Numerical Linear Algebra; Application of Methods of Numerical Linear Algebra in Science, Engineering and Economics.