A Bregman firmly nonexpansive proximal operator for baryconvex optimization

We present a generalization of the proximal operator defined through a convex combination of convex objectives, where the coefficients are updated in a minimax fashion. We prove that this new operator is Bregman firmly nonexpansive with respect to a Bregman divergence that combines Euclidean and information geometries. In this article we present a generalization of the convex optimization formalism (Boyd and Vandenberghe [2004]) that we call baryconvex optimization since it involves weighted convex objectives where the weights are learned in a minimax fashion. This paper proposes to extend well-known convex optimization methods such as the proximal point algorithm (PPA, see Rockafellar [1976]) and gradient descent (GD, see Boyd and Vandenberghe [2004]) to our general setting with S 1. Question: Can we compute a fixed point (if it exists) of the generalized prox in Definition 1? As will be shown, the answer provided by this paper is positive.