Convergence of the Robbins-Monro method for linear problems in a Banach space
Consider the equation \(Ax=b\), where b belongs to some Banach space X and A is a bounded linear operator from X into itself. It has been noticed by several people under different restrictions that in order to obtain an iterative solution one can use the Robbins-Monro method. The recursive formula is given by \(x_{n+1}=x_ n-\alpha_ n\quad (A_ nx_ n-b_ n),\) common assumptions are \(\alpha_ n\to 0,\) \(\sum \alpha_ n=\infty\) and the weighted means of \(A_ n\) and \(b_ n\) converge to A and b, respectively. The aim of the authors is to give a unified treatment of the convergence of \((x_ n)\) to a solution of \(Ax=b\) under mild conditions.
scientific article; zbMATH DE number 4119437
scientific article