Optimal scaling of the memoryless quasi-Newton updating formulas

Matrix approximations generated by the quasi-Newton (QN) updates may be generally vulnerable to ill-conditioning. Thus, the QN algorithms for unconstrained optimization may fail to suggest a proper trajectory to the solution. Here, by matrix analyses, it is discussed that how the classic scaling schemes of the QN algorithms can be modified to make further improvement in the computational stability of the methods. The argument mainly centers on a well-know open problem.