7 research outputs found
A weakly convergent fully inexact Douglas-Rachford method with relative error tolerance
Douglas-Rachford method is a splitting algorithm for finding a zero of the
sum of two maximal monotone operators. Each of its iterations requires the
sequential solution of two proximal subproblems. The aim of this work is to
present a fully inexact version of Douglas-Rachford method wherein both
proximal subproblems are solved approximately within a relative error
tolerance. We also present a semi-inexact variant in which the first subproblem
is solved exactly and the second one inexactly. We prove that both methods
generate sequences weakly convergent to the solution of the underlying
inclusion problem, if any
Projection-proximal methods for general variational inequalities
AbstractIn this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems
Modified extragradient methods for solving variational inequalities
AbstractIn this paper, we propose two methods for solving variational inequalities. In the first method, we modified the extragradient method by using a new step size while the second method can be viewed as an extension of the first one by performing an additional projection step at each iteration and another optimal step length is employed to reach substantial progress in each iteration. Under certain conditions, the global convergence of two methods is proved. Preliminary numerical experiments are included to illustrate the efficiency of the proposed methods