## 1 Introduction
This results in idle times, both for the master and for workers who respond quickly [10]. One way to reduce idle times is to use asynchronous computing (ASYN). In contrast to SYN, in ASYN, there are no synchronization points, so the master and workers do not need to wait until all workers respond to continue their operations. Thus, in an iterative process, e.g., in BD, the master would compute the next iterate based on information of possibly only a proper, but nonempty, subset of the workers. Based on this possibly incomplete information, the master sends a new iterate to available workers, while slower workers are still carrying their tasks on an outdated iterate. Because in ASYN iterates might not be evaluated by all workers, the evaluation of the objective function (yielding bounds on the optimal values) is precluded. Hence, a fundamental step in ASYN is the (scarce) coordination of workers to produce valid bounds. ASYN implementations have been proposed in the UC literature mainly to solve the dual problems (issued by LRs) via either subgradient algorithms or cutting-plane-based methods [15; 16; 17]. In References [15; 16], a queue of iterates is created and its elements are gradually sent to the workers. Auxiliary lists keep track of the evaluation status of each worker with respect tothe elements in the queue. Once an element has been evaluated by all workers, a valid bound to the original problem is available. The authors of Reference [15] demonstrate that their algorithm converges to a dual global solution regardless of the iterate-selection policy used to choose the iterates from the queue--first-in-first-out or last-in-first-out. In References [17], the authors keep a list of all the iteratesto compute valid bounds. In addition to solving the dual problem asynchronously, Reference [17] also conducts the primal recovery asynchronously. While References [15; 16] employ a convex trust-region bundle method, Reference [17] implements an incremental subgradient method. Asynchronous implementations of BD for convex problems can be found in References [18; 19; 20]. In Reference [18], the dual dynamic-programming algorithm is handled asynchronously in a hydrothermal scheduling problem. In Reference [19], the stochastic dual dynamic-programming algorithm is used for addressing the long-term planning problem of a hydro-dominated system: The authors propose to compute Benders cuts in an asynchronous fashion. This is also the case in Reference [20], where the authors consider an asynchronous Benders decomposition for convex multistage stochastic programming. Despite being successfully applied in a variety of fields, e.g., References [18; 19] and the references in References [21], the classical BD is well-known to suffer from slow convergence due to the oscillatory nature of [PERSON]'s cutting-plane method [22; 23]. Regularized BDs have been proven to outperform the classical one in several problems: See Reference [24] for (convex) two-stage linear programming, Reference [25] for (nonconvex) chance-constrained problems, and Reference [26] for robust designed of stations in water distribution networks. Several types of regularization exist [25; 27; 28]: proximal, trust-region, and level sets. Among the regularization methods, the level bundle method [29], also known as level decomposition (LD) in two-stage programming [24], stands out for its flexibility in dealing with convex or nonconvex feasible sets, stability functions and centers, and inexact oracles [25; 26; 30]. Recently, asymptotically level bundle methods for convex optimization were proposed in Reference [31]. The paper presents two algorithms. The first one does not employ coordination, but it makes use of upper bounds on the Lipschitz constants of the involved functions to compute upper bounds for the problem. The second algorithm does not make use of the latter assumption but requires scarce coordination.