Further Conjugation Results

Abstract

In this chapter, we exhibit several deeper results on conjugation. We first discuss Moreau’s decomposition principle, whereby a vector is decomposed in terms of the proximity operator of a lower semicontinuous function and that of its conjugate. This powerful nonlinear principle extends the standard linear decomposition with respect to a closed linear subspace and its orthogonal complement. Basic results concerning the proximal average and positively homogeneous functions are also presented. Also discussed are the Moreau– Rockafellar theorem, which characterizes coercivity in terms of an interiority condition, and the Toland–Singer theorem, which provides an appealing formula for the conjugate of the difference.