Composition duality and maximal monotonicity

This paper shows how the Attouch–Théra duality principle for operator inclusions can be extended to compositions of multifunctions, so that the primal and dual inclusions may involve operators between different pairs of spaces. We first present the extension and give an example from computational economics to demonstrate that it has practical utility. Then we consider a particular case, often found in applications, involving real Hilbert spaces and maximal monotone operators. We show that in this case our duality framework includes those previously developed by Rockafellar, Mosco, and Gabay. Finally we demonstrate that in this case, under very simple hypotheses the duality transformation preserves the maximal monotonicity of the operators involved. This proof uses an apparently new criterion for maximal monotonicity of operators of the form L*TL, where T is maximal monotone and L is linear and continuous with adjoint L*.