No abstract

The projection onto the intersection of sets generally does not allow for a closed form even when the individual projection operators have explicit descriptions. In this work, we systematically analyze the projection onto the intersection of a cone with either a ball or a sphere. Several cases are provided where the projector is available in closed form. Various examples based on finitely generated cones, the Lorentz cone, and the cone of positive semidefinite matrices are presented. The usefulness of our formulae is illustrated by numerical experiments for determining copositivity of real symmetric matrices.2010 Mathematics Subject Classification: Primary 47H09, 52A05, 90C25, 90C26; Secondary 15B48, 47H04.

Optical networks constitute a fundamental building block that has enabled the success of cloud computing. Virtualization, a cornerstone of cloud computing, today is applied in the networking field: physical network infrastructure is logically partitioned into separate virtual networks, thus providing isolation between distinct virtual network operators (VNOs). Hence, the problem of virtual network mapping has arisen: how to decide which physical resources to allocate for a particular virtual network? In a cloud context, not just network connectivity is required, but also data center (DC) resources located at multiple locations, for computation and/or storage. Given the underlying anycast routing principle, the network operator has some freedom to which specific DC to allocate these resources.In this paper, we solve a resilient virtual network mapping problem that optimally decides on the mapping of both network and multi-location data center resources resiliently using anycast routing, considering time-varying traffic conditions. In terms of resilience, we consider the so-called VNO-resilience scheme, where resilience is provided in the virtual network layer. To minimize physical resource capacity requirements, we allow reuse of both network and DC resources. The failures we protect against include both network and DC resource failures: we hence allocate backup DC resources, and also account for synchronization between primary and backup DC.As optimization criteria, we not only consider resource usage minimization, but also aim to limit virtual network reconfigurations from one time period to the next. We propose a scalable column generation approach to solve the dynamic resilient virtual network mapping problem, and demonstrate it in a case study on a nationwide US backbone network.

No abstract

We propose an iterative method for finding a zero of the sum of two maximally monotone operators in reflexive Banach spaces. One of the operators is single-valued, and the method alternates an explicit step on this operator and an implicit step on the other one. Both steps involve the gradient of a convex function that is free to vary over the iterations. The convergence of the resulting forward-backward splitting method is analyzed using the theory of Legendre functions, under a novel assumption on the single-valued operator that captures various existing properties. When applied to minimization problems, rates are obtained for the objective values. The proposed framework unifies and extends several iterative methods which have thus far not been brought together, and it is also new in Euclidean spaces.