一种新的修正SR1更新公式及其算法收敛性

SR1更新公式对比其他的拟牛顿更新公式，会更加简单且每次迭代需要更少的计算量。但是一般SR1更新公式的收敛性质是在一致线性无关这一很强的条件下证明的。基于前人的研究成果，提出了一种新的修正SR1公式，并分别证明了其在一致线性无关和没有一致线性无关这两个条件下的局部收敛性，最后通过数值实验验证了提出的更新公式的有效性，以及所作出假设的合理性。根据实验数据显示，在某些条件下基于所提出更新公式的拟牛顿算法会比基于传统的SR1更新公式的算法收敛效果更好一些。

A new modified SR1 update formulas and its algorithm convergence

The Quasi-Newton method is widely regarded as one of the most effective methods to solve the problem of small scale optimization problem. They avoid the problem of solving the hession matrix by Newton method. In fact, the Quasi-newton method produces a symmetric matrix to approximate the hession matrix at each iteration. There are many Quasi-newton update formulas, frequently, such as BFGS update formula, SR1 update formula, DFP update formula and so on. Compared to other Quasi-Newton update formulas, SR1 Update formulas are more simpler and need less calculation. But most of time its convergence is improved on the strict condition of uniformly linearly independent. So this paper proposes a new modified SR1 Update formalus, and analyzes the convergence of the quasi-Newton algorithm based on the new modified formula SR1 under two different hypothesises respectively, including the condition of uniformly linearly independent and not uniformly linearly independent. By avoiding uniformly linearly independent and adding the assumption of positive-definite bounded hessian approximations, the new modified SR1 Update formalus is proved to be $n$+1 step $q$-superlinearly convergent. In addition, We also validate the reasonableness of assumptions and effectiveness of algorithms through experiments. The numerical results are reported to support that new update formulas proposed in the paper has better convergence rate in some conditions.