A memoryless symmetric rank-one method with sufficient descent property for unconstrained optimization

Abstract

Quasi-Newton methods are widely used for solving unconstrained optimization problems. However, it is difficult to apply quasi-Newton methods directly to large-scale unconstrained optimization problems, because they need the storage of memories for matrices. In order to overcome this difficulty, memoryless quasi-Newton methods were proposed. Shanno (1978) derived the memoryless BFGS method. Recently, several researchers studied the memoryless quasi-Newton method based on the symmetric rank-one formula. However existing memoryless symmetric rank-one methods do not necessarily satisfy the sufficient descent condition. In this paper, we focus on the symmetric rank-one formula based on the spectral scaling secant condition and derive a memoryless quasi-Newton method based on the formula. Moreover we show that the method always satisfies the sufficient descent condition and converges globally for general objective functions. Finally, preliminary numerical results are shown.