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

Abstract In this paper, combining the new splitting method proposed by Eckstein and Svaiter and the forcing strong convergence method of Solodov and Svaiter, we propose a splitting proximal projection method for the sum of two maximal monotone operators. We prove that the proposed method is well-defined whether the solution set of the problem is nonempty or not and the sequence generated by the method converges strongly to an extended solution of the problem.

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

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

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

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

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

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

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

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

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

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

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

[12]  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..

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

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

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

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