Variational approximation of functionals defined on 1-dimensional connected sets: the planar case

In this paper we consider variational problems involving 1-dimensional connected sets in the Euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica-Mortola type energies proving a $\Gamma$-convergence result. We also introduce a suitable convex relaxation and develop the corresponding numerical implementations. The proposed methods are quite general and the results we obtain can be extended to $n$-dimensional Euclidean space or to more general manifold ambients, as shown in the companion paper [11].Comment: 30 pages, 5 figure

Performance improvement of an optical network providing services based on multicast

Operators of networks covering large areas are confronted with demands from some of their customers who are virtual service providers. These providers may call for the connectivity service which fulfils the specificity of their services, for instance a multicast transition with allocated bandwidth. On the other hand, network operators want to make profit by trading the connectivity service of requested quality to their customers and to limit their infrastructure investments (or do not invest anything at all). We focus on circuit switching optical networks and work on repetitive multicast demands whose source and destinations are {\em \`a priori} known by an operator. He may therefore have corresponding trees "ready to be allocated" and adapt his network infrastructure according to these recurrent transmissions. This adjustment consists in setting available branching routers in the selected nodes of a predefined tree. The branching nodes are opto-electronic nodes which are able to duplicate data and retransmit it in several directions. These nodes are, however, more expensive and more energy consuming than transparent ones. In this paper we are interested in the choice of nodes of a multicast tree where the limited number of branching routers should be located in order to minimize the amount of required bandwidth. After formally stating the problem we solve it by proposing a polynomial algorithm whose optimality we prove. We perform exhaustive computations to show an operator gain obtained by using our algorithm. These computations are made for different methods of the multicast tree construction. We conclude by giving dimensioning guidelines and outline our further work.Comment: 16 pages, 13 figures, extended version from Conference ISCIS 201

Partitions of Minimal Length on Manifolds

We study partitions on three dimensional manifolds which minimize the total geodesic perimeter. We propose a relaxed framework based on a $\Gamma$-convergence result and we show some numerical results. We compare our results to those already present in the literature in the case of the sphere. For general surfaces we provide an optimization algorithm on meshes which can give a good approximation of the optimal cost, starting from the results obtained using the relaxed formulation

Calibrations for minimal networks in a covering space setting

In this paper we define a notion of calibration for an equivalent approach to the classical Steiner problem in a covering space setting and we give some explicit examples. Moreover we introduce the notion of calibration in families: the idea is to divide the set of competitors in a suitable way, defining an appropriate (and weaker) notion of calibration. Then, calibrating the candidate minimizers in each family and comparing their perimeter, it is possible to find the minimizers of the minimization problem. Thanks to this procedure we prove the minimality of the Steiner configurations spanning the vertices of a regular hexagon and of a regular pentagon

Distributed Optimization With Local Domains: Applications in MPC and Network Flows

In this paper we consider a network with $P$ nodes, where each node has exclusive access to a local cost function. Our contribution is a communication-efficient distributed algorithm that finds a vector $x^\star$ minimizing the sum of all the functions. We make the additional assumption that the functions have intersecting local domains, i.e., each function depends only on some components of the variable. Consequently, each node is interested in knowing only some components of $x^\star$, not the entire vector. This allows for improvement in communication-efficiency. We apply our algorithm to model predictive control (MPC) and to network flow problems and show, through experiments on large networks, that our proposed algorithm requires less communications to converge than prior algorithms.Comment: Submitted to IEEE Trans. Aut. Contro

Dual Averaging Method for Online Graph-structured Sparsity

Online learning algorithms update models via one sample per iteration, thus efficient to process large-scale datasets and useful to detect malicious events for social benefits, such as disease outbreak and traffic congestion on the fly. However, existing algorithms for graph-structured models focused on the offline setting and the least square loss, incapable for online setting, while methods designed for online setting cannot be directly applied to the problem of complex (usually non-convex) graph-structured sparsity model. To address these limitations, in this paper we propose a new algorithm for graph-structured sparsity constraint problems under online setting, which we call \textsc{GraphDA}. The key part in \textsc{GraphDA} is to project both averaging gradient (in dual space) and primal variables (in primal space) onto lower dimensional subspaces, thus capturing the graph-structured sparsity effectively. Furthermore, the objective functions assumed here are generally convex so as to handle different losses for online learning settings. To the best of our knowledge, \textsc{GraphDA} is the first online learning algorithm for graph-structure constrained optimization problems. To validate our method, we conduct extensive experiments on both benchmark graph and real-world graph datasets. Our experiment results show that, compared to other baseline methods, \textsc{GraphDA} not only improves classification performance, but also successfully captures graph-structured features more effectively, hence stronger interpretability.Comment: 11 pages, 14 figure