# Zeros of Sums of Monotone Operators

## Abstract

Properties of the zeros of a single monotone operator were discussed in Section 23.4. In this chapter, we first characterize the zeros of sums of monotone operators and then present basic algorithms to construct such zeros iteratively. Duality for monotone inclusion problems is also discussed. In the case of two operators *A* and *B* such that *A* + *B* is maximally monotone, a point in \({\operatorname {zer}}(A+B)\) could in principle be constructed via Theorem 23.41. However, this approach is numerically viable only when it is easy to compute *J*_{γ(A+B)}. A more widely applicable alternative is to devise an *operator splitting* algorithm, in which the operators *A* and *B* are employed in separate steps. Various such algorithms are discussed in this chapter.

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