To minimize a convex function [??], we state a class of penalty-type bundle algorithms, where the penalty uses a variable metric. This metric is updated according to quasi-Newton formulae based on Moreau-Yosida approximations of [??]. In particular, we introduce a "reversal" quasi-Newton formula, specially suited for our purpose. We consider several variants in the algorithm and discuss their respective merits. Furthermore, we accept a degenerate penalty term in the Moreau-Yosida regularization.