Generalized Bundle Methods
研究一类广义束方法，其中稳定项可以是满足特定性质的任意闭凸函数，覆盖了文献中多种算法，并证明了有限终止、渐近收敛和有限收敛到最优点的性质。
We study a class of generalized bundle methods for which the stabilizing term can be any closed convex function satisfying certain properties. This setting covers several algorithms from the literature that have been so far regarded as distinct. Under a different hypothesis on the stabilizing term and/or the function to be minimized, we prove finite termination, asymptotic convergence, and finite convergence to an optimal point, with or without limits on the number of serious steps and/or requiring the proximal parameter to go to infinity. The convergence proofs leave a high degree of freedom in the crucial implementative features of the algorithm, i.e., the management of the bundle of subgradients ($\beta$-strategy) and of the proximal parameter (t-strategy). We extensively exploit a dual view of bundle methods, which are shown to be a dual ascent approach to one nonlinear problem in an appropriate dual space, where nonlinear subproblems are approximately solved at each step with an inner linearization approach. This allows us to precisely characterize the changes in the subproblems during the serious steps, since the dual problem is not tied to the local concept of $\varepsilon$-subdifferential. For some of the proofs, a generalization of inf-compactness, called *-compactness, is required; this concept is related to that of asymptotically well-behaved functions.