Warped Proximal Iterations for Monotone Inclusions

Resolvents play a central role in the design and the analysis of splitting algorithms for solving monotone inclusions. We investigate a generalization of this notion, called warped resolvent, which is constructed with the help of an auxiliary operator. Iterations involving warped resolvents are shown to capture a wide range of algorithmic schemes and to lead to new monotone operator splitting methods. Particular attention is given to cutting plane algorithms, which operate with successive outer approximations of the solution set involving one or two affine half-spaces. Weakly and strongly convergent algorithms are devised in this context.