On Bauschke-Bendit-Moursi modulus of averagedness and generalized monotone operators

Averaged operators are important in convex analysis and optimization algorithms. In the first part of this thesis, we propose classifications of averaged operators, firmly nonexpansive operators, and proximal operators by using the Bauschke-Bendit-Moursi modulus of averagedness. We show that if an averaged operator has modulus of averagedness less than 1/2, then it is a bi-Lipschitz homeomorphism. Amazingly the proximity operator of a convex function has its modulus of averagedness less than 1/2 if and only if the function is Lipschitz smooth. Some results on the averagedness of operator compositions are obtained. Explicit formulae for calculating the modulus of averagedness of resolvents and proximity operators in terms of various values associated with the maximally monotone operator or subdifferential are also given. Examples are provided to illustrate our results. In the second part of this thesis, we focus on a natural generalization of averaged operators and monotonicity. We extend the definition of modulus of averagedness and show how it behaves for linear operators. As an application, explicit formulae are obtained with respect to angle between subspaces. We will also present various profound nonlinear results about this generalization of averaged operator and monotonicity. All the results can apply to averaged operators. Examples are provided to illustrate our results.