Optimal scaling of the memoryless quasi–Newton updatingformulas

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