10. Proximal Algorithms#

10.1. Chapter Objectives#

  • Proximal mappings

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

  • Proximal operators

10.2. 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.