Convergence analysis of a method for variational inclusions

Abstract

Consider the following variational inclusion problem:

where f is differentiable in a neighbourhood of a solution and g is differentiable at , and F is a set-valued mapping, and the method introduced in Jean-Alexis and Pietrus [C. Jean-Alexis and A. Pietrus, On the convergence of some methods for variational inclusions, Rev. R. Acad. Cien. serie A. Mat. 102(2) (2008), pp. 355–361] for solving this problem:

where ∇f(x) denotes the Fréchet derivative of f at x and [x, y; g] the first-order divided difference of g on the points x and y. Local converge analysis are provided for the method under the weaker conditions than Jean-Alexis and Pietrus (2008). Moreover, if ∇f and the first-order divided difference of g are p-Hölder continuous at a solution, then we show that this method converges superlinearly. In particular, our results extend the corresponding ones Jean-Alexis and Pietrus (2008), and fix a gap in the proof in (Jean-Alexis and Pietrus (2008), Theorem 1).

Keywords:

• set-valued mapping
• variational inclusions
• pseudo-Lipschitz continuity
• local convergence
• divided difference

AMS Subject Classifications::

• Primary: 47H04, 90C30
• Secondary: 49J53, 65K10

Acknowledgements

M.H. Rashid was fully supported by China Scholarship Council. Research of J.H. Wang was partially supported by the National Natural Science Foundation of China (grant 11001241). C. Li was supported in part by the National Natural Science Foundation of China (grant 11171300) and by Zhejiang Provincial Natural Science Foundation of China (grant Y6110006).