Strong convergence of a splitting projection method for the sum of maximal monotone operators

In this paper, a projective-splitting method is proposed for finding a zero of the sum of $$n$$ maximal monotone operators over a real Hilbert space $$\mathcal{H }$$. Without the condition that either $$\mathcal{H }$$ is finite dimensional or the sum of $$n$$ operators is maximal monotone, we prove that the sequence generated by the proposed method is strongly convergent to an extended solution for the problem, which is closest to the initial point. The main results presented in this paper generalize and improve some recent results in this topic.

[1]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[2]  P. Lions,et al.  Splitting Algorithms for the Sum of Two Nonlinear Operators , 1979 .

[3]  B. Svaiter,et al.  A strongly convergent hybrid proximal method in Banach spaces , 2004 .

[4]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

[5]  Michel Théra,et al.  Graphical convergence of sums of monotone mappings , 2002 .

[6]  H. Attouch.  A General Duality Principle for the Sum of Two Operators 1 , 1996 .

[7]  A. Moudafi,et al.  Convergence of a splitting inertial proximal method for monotone operators , 2003 .

[8]  Benar Fux Svaiter,et al.  Forcing strong convergence of proximal point iterations in a Hilbert space , 2000, Math. Program..

[9]  Benar Fux Svaiter,et al.  General Projective Splitting Methods for Sums of Maximal Monotone Operators , 2009, SIAM J. Control. Optim..

[10]  A. Fischer,et al.  A family of operator splitting methods revisited , 2010 .

[11]  Benar Fux Svaiter,et al.  A family of projective splitting methods for the sum of two maximal monotone operators , 2007, Math. Program..

[12]  R. Rockafellar.  On the maximality of sums of nonlinear monotone operators , 1970 .

[13]  Nan-Jing Huang,et al.  Strong convergence of a splitting proximal projection method for the sum of two maximal monotone operators , 2012, Oper. Res. Lett..

[14]  Benar Fux Svaiter,et al.  On Weak Convergence of the Douglas-Rachford Method , 2010, SIAM J. Control. Optim..

[15]  G. Minty.  Monotone (nonlinear) operators in Hilbert space , 1962 .

[16]  Heinz H. Bauschke.  A Note on the Paper by Eckstein and Svaiter on "General Projective Splitting Methods for Sums of Maximal Monotone Operators" , 2009, SIAM J. Control. Optim..

[17]  Naihua Xiu,et al.  Modified Extragradient Method for Variational Inequalities and Verification of Solution Existence , 2003 .

[18]  J. Mashreghi,et al.  Forcing strong convergence of Korpelevich’s method in Banach spaces with its applications in game theory , 2010 .

[19]  Abdellatif Moudafi,et al.  On the regularization of the sum of two maximal monotone operators , 2000 .