Modified approximate proximal point algorithms for finding roots of maximal monotone operators

Abstract

In order to find roots of maximal monotone operators, this paper introduces and studies the modified approximate proximal point algorithm with an error sequence {e ^k} such that $\left\| { e^k } \right\| \leqslant \eta _k \left\| { x^k - \tilde x^k } \right\|$ with $\sum\limits_{k = 0}^\infty {\left( {\eta _k - 1} \right)} < + \infty$ and $\mathop {\inf }\limits_{k \geqslant 0} \eta _k = \mu \geqslant 1$. Here, the restrictions on {η _k} are very different from the ones on {η _k}, given by He et al (Science in China Ser. A, 2002, 32 (11): 1026–1032.) that $\mathop {\sup }\limits_{k \geqslant 0} \eta _k = v < 1$. Moreover, the characteristic conditions of the convergence of the modified approximate proximal point algorithm are presented by virtue of the new technique very different from the ones given by He et al.