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.
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.
Beck and Teboulle's FISTA method for finding a minimizer of the sum of two convex functions, one of which has a Lipschitz continuous gradient whereas the other may be nonsmooth, is arguably the most important optimization algorithm of the past decade. While research activity on FISTA has exploded ever since, the mathematically challenging case when the original optimization problem has no minimizer has found only limited attention.In this work, we systematically study FISTA and its variants. We present general results that are applicable, regardless of the existence of minimizers. Proof. See Appendix A.Lemma 2.2 Let (α n ) n∈N * and (β n ) n∈N * be sequences in R + . Suppose that ∑ n∈N * α n = +∞ and that ∑ n∈N * α n β n < +∞. Then lim β n = 0.Proof. See Appendix B.The novelty of the following result lies in the fact that the error sequence (ε n ) n∈N * need not lie in R + .
Multi-layer optical networks have recently evolved towards IP-over-WDM networks. Therein, in order to avoid protection/restoration redundancies against either single or multiple failures, synergies need to be developed between IP and optical layers in order to reduce the costs and the energy consumption of the future IP-over-WDM networks.We propose two new optimization models. The first one is an enhanced cutset model, relying on a column generation reformulation. The second one is a path model, based on a multiflow formulation. Both models can solve exactly most benchmark instances, which were only solved heuristically so far. I. INTRODUCTIONThe design and the management of the future networks will rely on an all IP-design, where synergies will need to be developed between the IP and the optical layers in order to reduce the energy consumption and the network costs, as well as to guarantee the Service Level Agreements (SLA) while bandwidth greedy applications, like video services and IPTV services, will continue to grow.Network failures, such as link or node failures, cannot be fully avoided when it comes to network management. Consequently, a backup mechanism needs to be used to ensure the network connectivity. When a failure occurs, the backup mechanism establishes an alternative path to carry the interrupted connections. Depending on whether this alternative path is generated online or offline, the corresponding backup mechanism is referred to as restoration or protection, respectively. In multi-layer networks, failures, whether single or multiple ones, have been traditionally taken care by the layer in which they appear, throughout restoration mechanisms in the IP layer and protection or restoration ones in the optical layer. In order to enhance the efficiency of the future networks, with respect to costs, energy consumption and quality of service, cross-layer protection/restoration is now more often envisionedThe IP layer is referred to as the logical/virtual layer where each logical link is mapped to a lightpath (i.e., a direct optical connection without any intermediate electronics) in the optical/physical layer. A single duct failure in an optical network can result in several broken physical links, and consequently in several disrupted logical links, and in a disconnected logical topology. Hence, a first condition for the existence of a survivable logical topology is to remain connected (survivable) in case of any network failure [3], and then to make sure enough bandwidth is available in order to ensure successful IP restorations. However, synergies need to be developed in
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