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 $f:\mathbb{R}\to (-\mathrm{\infty },\mathrm{\infty }]$:

• If $f^{\prime }(u)=0$, 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.