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We consider the incremental subgradient algorithm employing dynamic step sizes for minimizing the sum of a large number of component convex functions. The dynamic step size rule was firstly introduced by Goffin and Kiwiel [Math. Program., 1999, 85(1): 207‐211] for the subgradient algorithm, soon later, for the incremental subgradient algorithm by Nedic and Bertsekas in [SIAM J. Optim., 2001, 12(1): 109‐138]. It was observed experimentally that the incremental approach has been very successful in solving large separable optimizations and that the dynamic step sizes generally have better computational performance than others in the literature. In the present paper, we propose two modified dynamic step size rules for the incremental subgradient algorithm and analyse the convergence and complexity properties of them. At last, the assignment problem is considered and the incremental subgradient algorithms employing different kinds of dynamic step sizes are applied to solve the problem. The computational experiments show that the two modified ones converges dramatically faster and more stable than the corresponding one in [SIAM J. Optim., 2001, 12(1): 109‐138]. Particularly, for solving large separable convex optimizations, we strongly recommend the second one (see Algorithm 3.3 in the paper) since it has interesting computational performance and is the simplest one. [ABSTRACT FROM AUTHOR]

Summary: Recently, the routing problem in vehicular ad hoc networks is one of the most vital research. Despite the variety of the proposed approaches and the development of communications technologies, the routing problem in VANET suffers from the high speed of vehicles and the repetitive failures in communications. In this paper, we adjusted the well‐known K‐medoids clustering algorithm to improve the network stability and to increase the lifetime of all established links. First, the number of clusters and the initial cluster heads will not be selected randomly as usual, but based on mathematical formula considering the environment size and the available transmission ranges. Then the assignment of nodes to clusters in both k‐medoids phases will be carried out according to several metrics including direction, relative speed, and proximity. To the best of our knowledge, our proposed model is the first that introduces the new metric named "node disconnection frequency." This metric prevents nodes with volatile and suspicious behavior to be elected as a new CH. This screening ensures that the new CH retains its property as long as possible and thus increases the network stability. Empirical results confirm that in addition to the convergence speed that characterizes our adjusted K‐medoids clustering algorithm (AKCA), the proposed model achieves more stability and robustness when compared with most recent approaches designed for the same objective. [ABSTRACT FROM AUTHOR]