In this work we introduce and study novel Quasi Newton minimization methods based on a Hessian approximation Broyden Class-type updating scheme, where a suitable matrix \(\tilde{B}_k\) is updated instead of the current Hessian approximation \(B_k\). We identify conditions which imply the convergence of the algorithm and, if exact line search is chosen, its quadratic termination. By a remarkable connection between the projection operation and Krylov spaces, such conditions can be ensured using low complexity matrices \(\tilde{B}_k\) obtained projecting \(B_k\) onto algebras of matrices diagonalized by products of two or three Householder matrices adaptively chosen step by step. Experimental tests show that the introduction of the adaptive criterion, which theoretically guarantees the convergence, considerably improves the robustness of the minimization schemes when compared with a non-adaptive choice; moreover, they show that the proposed methods could be particularly suitable to solve large scale problems where L-BFGS is not able to deliver satisfactory performance.

中文翻译：

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使用Householder自适应变换的Broyden类方法的一种变体

在这项工作中，我们介绍和研究基于Hessian逼近Broyden类类型更新方案的新颖的拟牛顿最小化方法，其中更新了合适的矩阵\（\ tilde {B} _k \）而不是当前的Hessian逼近\（B_k \ ）。我们确定条件，这些条件暗示算法的收敛性，如果选择精确的线搜索，则其二次终止。通过投影操作和Krylov空间之间的显着联系，可以使用获得的投影\（B_k \）的低复杂度矩阵\（\ tilde {B} _k \）来确保这样的条件逐步地将两个或三个Householder矩阵的乘积对角化到矩阵的代数上。实验测试表明，与非自适应选择相比，从理论上保证收敛的自适应准则的引入大大提高了最小化方案的鲁棒性。此外，他们表明，所提出的方法可能特别适合解决L - BFGS无法提供令人满意的性能的大规模问题。