The Generalized Bregman Distance

Recently, a new kind of distance has been introduced for the graphs two point-to-set operators, one which is maximally monotone. When both operators are subdifferential proper lower semicontinuous convex function, this specializes under modest assumptions to classical Bregman distance. We name generalized distance, and we shed light on it with examples that utilize other most natural representative functions: Fitzpatrick function its conjugate. provide sufficient conditions convexity, coercivity, supercoercivity: properties essential implementation in proximal point type algorithms. establish these results left right variants construct closely related Kullback--Leibler divergence, was previously considered context distances whose importance information theory well known. In so doing, demonstrate how compute difficult conjugate discover occurrences Lambert ${\mathcal W}$ optimization growing interest.