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Within convex analysis, a rich theory with various applications has been evolving since the proximal average of convex functions was first introduced over a decade ago. When one considers the subdifferential of the proximal average, a natural averaging operation of the subdifferentials of the averaged functions emerges. In the present paper we extend the reach of this averaging operation to the framework of monotone operator theory in Hilbert spaces, transforming it into the resolvent average. The theory of resolvent averages contains many desirable properties. In particular, we study a detailed list of properties of monotone operators and classify them as dominant or recessive with respect to the resolvent average. As a consequence, we recover a significant part of the theory of proximal averages. Furthermore, we shed new light on the proximal average and present novel results and desirable properties the proximal average possesses which have not been previously available.

In this paper, we consider a conical extension of averaged nonexpansive operators and its usefulness in the convergence analysis of fixed point algorithms. Various properties of conically averaged operators are systematically studied, in particular, the stability under relaxations, convex combinations, and compositions. We then derive the conical averagedness of the resolvents of operators that possess certain types of generalized monotonicity. These properties are used to analyze the convergence of fixed point algorithms including the proximal point algorithm, the forward-backward algorithm, and the adaptive Douglas-Rachford algorithm. Our results not only unify and improve recent studies but also provide a different perspective on the topic.

We study a conical extension of averaged nonexpansive operators and the role it plays in convergence analysis of fixed point algorithms. Various properties of conically averaged operators are systematically investigated, in particular, the stability under relaxations, convex combinations and compositions. We derive conical averagedness properties of resolvents of generalized monotone operators. These properties are then utilized in order to analyze the convergence of the proximal point algorithm, the forward-backward algorithm, and the adaptive Douglas-Rachford algorithm. Our study unifies, improves and casts new light on recent studies of these topics.

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