Asymptotic Convergence Analysis of The Proximal Point Algorithm for Metrically Regular Mappings

Matsushita Shin-ya

Xu Li

5th International Workshop on Computational Intelligence & Applications Proceedings : IWCIA 2009

This paper studies convergence properties of the proximal point algorithm when applied to a certain class of nonmonotone set-valued mappings. We consider an algorithm for solving an inclusion 0 ∈ T(x), where T is a metrically regular set-valued mapping acting from Rn into Rm. The algorithm is given by the follwoing iteration: x0 ∈ Rn and

xk+1 = αkxk + (1 - αk)yk, for k = 0, 1, 2, . . .,

where {αk} is a sequence in [0, 1] such that αk ≤ ¯α < 1, gk is a Lipschitz mapping from Rn into Rm and yk satisfies the following inclusion

0 ∈ gk(yk) - gk(xk) + T(yk).

We prove that if the modulus of regularity of T is sufficiently small then the sequence generated by our algorithm converges to a solution to 0 ∈ T(x).

Technology. Engineering [ 500 ]

eng

IEEE SMC Hiroshima Chapter

(c) Copyright by IEEE SMC Hiroshima Chapter.

[ISSN] 1883-3977

[URI] http://www.hil.hiroshima-u.ac.jp/iwcia/2009/