On Borwein–Wiersma Decompositions of Monotone Linear Relations

Abstract

Monotone operators are of basic importance in optimization as they generalize simultaneously subdifferential operators of convex functions and positive semidefinite (not necessarily symmetric) matrices. In 1970, Asplund [A monotone convergence theorem for sequences of nonlinear mapping, in Nonlinear Functional Analysis, American Math Society, Providence, RI, 1970, pp. 1–9] studied the additive decomposition of a maximal monotone operator as the sum of a subdifferential operator and an “irreducible” monotone operator. In 2007, Borwein and Wiersma [SIAM J. Optim., 18 (2007), pp. 946–960] introduced another additive decomposition, where the maximal monotone operator is written as the sum of a subdifferential operator and a “skew” monotone operator. Both decompositions are variants of the well-known additive decomposition of a matrix via its symmetric and skew part. This paper presents a detailed study of the Borwein–Wiersma decomposition of a maximal monotone linear relation. We give sufficient conditions and characterizations for a maximal monotone linear relation to be Borwein–Wiersma decomposable and show that Borwein–Wiersma decomposability implies Asplund decomposability. We exhibit irreducible linear maximal monotone operators without full domain, thus answering one of the questions raised by Borwein and Wiersma. The Borwein–Wiersma decomposition of any maximal monotone linear relation is made quite explicit in Hilbert space.