The forward-backward algorithm and the normal problem
Theoretical contribution for researchers working on monotone inclusions and optimization algorithms, but incremental as it extends known results to the unsolvable case.
The paper studies the forward-backward algorithm when the monotone inclusion problem has no solution, providing a new formula for normal solutions and the range of the displacement map, illustrated with examples.
The forward-backward splitting technique is a popular method for solving monotone inclusions that has applications in optimization. In this paper we explore the behaviour of the algorithm when the inclusion problem has no solution. We present a new formula to define the normal solutions using the forward-backward operator. We also provide a formula for the range of the displacement map of the forward-backward operator. Several examples illustrate our theory.