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Abstract: In this paper, we concentrate on solving a convex time space network flow problem with decomposable structures. We first describe the convex time space network flow optimization model, and transform it into an equivalent variational inequality problem. Then, after exploring the decomposable structure of primal decision variables, we propose a novel decomposable self-adaptive projection-based prediction–correction algorithm (DSPPCA) to solve the model, and then further provide its convergent theory. Finally, we report the computational performances through computational experiments. Numerical results reveal that DSPPCA not only can enhance the accuracy and convergence rate significantly, but also can be a powerful search algorithm for convex optimization problems with decomposable structures of decision variables. [Copyright &y& Elsevier]

Abstract: It is known that a consensus problem on any connected complete Riemannian manifold can be transformed into the one on its strongly convex balls via the compression–decompression along geodesics. From the viewpoint of interior metrics, this paper mainly provides a consensus protocol for strongly convex geodesic balls, in which the communication can be switching and directed. With the aid of nonsmooth analysis tools on Riemannian manifolds, our analysis shows that all dynamical points involved can achieve consensus in finite time. Meanwhile, the corresponding global algorithm is given, with its application to the consensus problem of rotation attitudes, as well as a case simulation, to demonstrate and verify our proposed techniques. [Copyright &y& Elsevier]

Abstract: In view of the fact that prototypes of Riemannian manifolds range from Euclidean surfaces to Lie groups, operator spaces, etc., the paper explores common consensus schemes for connected complete Riemannian manifolds. After studying the restriction exponential map, we introduce a local consensus protocol with switching communication topologies for the manifolds via the isometric mapping, based on one of the Euclidean consensus protocols. Equipped with the compression–decompression functions, we extend the local consensus protocol, and put forth a global consensus algorithm for any such manifolds. These consensus schemes can provide common solutions to the consensus problems arising from various nonlinear spaces or abstract spaces belonging to connected complete Riemannian manifolds. In particular, they are simple and concise, and can converge in finite time. Simulation on the unit sphere is provided to verify our consensus schemes, and to demonstrate the usage of our proposed techniques. [Copyright &y& Elsevier]