  This paper presents a proximal bundle (PB) framework based on a generic
bundle update scheme for solving the hybrid convex composite optimization
(HCCO) problem and establishes a common iteration-complexity bound for any
variant belonging to it. As a consequence, iteration-complexity bounds for
three PB variants based on different bundle update schemes are obtained in the
HCCO context for the first time and in a unified manner. While two of the PB
variants are universal (i.e., their implementations do not require parameters
associated with the HCCO instance), the other newly (as far as the authors are
aware of) proposed one is not but has the advantage that it generates simple,
namely one-cut, bundle models. The paper also presents a universal adaptive PB
variant (which is not necessarily an instance of the framework) based on
one-cut models and shows that its iteration-complexity is the same as the two
aforementioned universal PB variants.
