12. Proximal Algorithms#
12.1. Chapter Objectives#
Existence and uniqueness of proximal mappings for proper, closed, convex functions
12.2. Relevant results#
We recall some results from previous chapters which will be helpful for the work in this chapter.
Sum of two closed functions is a closed function.
Some of a convex function with a strongly convex function is strongly convex.
A proper, closed and strongly convex function has a unique minimizer.
For some convex \(f: \RR \to \RERL\):
If \(f'(u) = 0\), then \(u\) must be one of its minimizers.
If the minimizer of \(f\) exists and is not attained at any point of differentiability, then it must be attained at a point of nondifferentiability.