# 12. Proximal Algorithms#

## 12.1. Chapter Objectives#

• Proximal mappings

• Existence and uniqueness of proximal mappings for proper, closed, convex functions

• Proximal operators

## 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.