In this work we introduce and study novel Quasi Newton minimization methods based on a Hessian approximation Broyden Class-\textit{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. Extended 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 $L$-$BFGS$ performs poorly.

中文翻译：

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使用 Householder 自适应变换的 Broyden 类方法的变体

在这项工作中，我们介绍并研究了基于 Hessian 近似 Broyden Class-\textit{type} 更新方案的新颖拟牛顿最小化方法，其中更新了合适的矩阵 $\tilde{B}_k$ 而不是当前的 Hessian 近似值 $B_k $. 我们确定暗示算法收敛的条件，如果选择精确的线搜索，则确定其二次终止。通过投影操作和 Krylov 空间之间的显着连接，可以使用低复杂度矩阵 $\tilde{B}_k$ 来确保这些条件，该矩阵将 $B_k$ 投影到由自适应选择步骤的两个或三个 Householder 矩阵的乘积对角化的矩阵代数上一步一步。扩展实验测试表明，自适应准则的引入，理论上保证了收敛，与非自适应选择相比，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 $L$-$BFGS$ 表现不佳的大规模问题。