A variation of Broyden class methods using Householder adaptive transforms

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 Btilde_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 Btilde_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.