The relationship between monotonicity and accretivity on Riemannian manifolds is studied in this paper and both concepts are proved to be equivalent in Hadamard manifolds. As a consequence an iterative method is obtained for approximating singularities of Lipschitz continuous, strongly monotone mappings.We also establish the equivalence between the strong convexity of convex functions and the strong monotonicity of its subdifferentials on Riemannian manifolds. These results are then applied to solve the minimization problem of convex functions on Riemannian manifolds.

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In the present paper, we study the finite termination of sequences generated by inexact proximal point algorithms for finding zeroes of a maximal monotone (set-valued) operator T on a Hilbert space. Under some mild conditions, we get that a sequence generated by inexact proximal point algorithm stops after a finite number of iterations. Our results extend the corresponding results in Rockafellar (SIAM J Control Optim 14:877-898, 1976). In particular, for optimization problems, our results improve corresponding results in Ferris (Math Progr 50:359-366, 1991). As applications, we obtain finite termination of projected gradient method.Communicated by Viorel Barbu.

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The k-shell decomposition plays an important role in unveiling the structural properties of a network, i.e., it is widely adopted to find the densest part of a network across a broad range of scientific fields, including Internet, biological networks, social networks etc. However, there arises concern about the robustness of the k-shell structure when networks suffer from adversarial attacks. Here, we introduce and formalize the problem of kshell attack and develop an efficient strategy to attack the k-shell structure by rewiring a small number of links. To the best of our knowledge, it is the first time to study the robustness of graph k-shell structure under adversarial attacks. In particular, we propose a Simulated Annealing (SA) based k-shell attack method and testify it on four real-world social networks. The extensive experiments validate that the k-shell structure of a network is robust under random perturbation, but it is quite vulnerable under adversarial attack, e.g., in Dolphin and Throne networks, more than 40% nodes change their k-shell values when only 10% links are changed based on our SA-based k-shell attack. Such results suggest that a single structural feature could also be significantly disturbed when only a small fraction of links are changed purposefully in a network. Therefore, it could be an interesting topic to improve the robustness of various network properties against adversarial attack in the future.