Identifying infeasible subsets of linear inequalities that are irreducible with respect to a given subset of the inequalities

A classical problem in the study of an infeasible system of linear inequalities is to determine irreducible infeasible subsets of inequalities (IIS), i.e. infeasible subsets of inequalities whose proper subsets are feasible. In this article, we examine a particular situation where only a given subsystem is of interest for the analysis of infeasibility. For this, we define B-IISs as infeasible subsets of inequalities that are irreducible with respect to a given subsystem. It is a generalization of the definition of an IIS, since an IIS is irreducible with respect to the full system. We provide a practical characterization of infeasible subsets irreducible with respect to a subsystem, making the link with the dual polytope commonly used in the detection of IISs. We then turn to the study of the BIISs that can be obtained from the Phase I of the simplex algorithm. We answer an open question regarding the covering of the clusters of such B-IISs and deduce a practical algorithm to find these covering B-IISs. Our findings are numerically illustratedon the Netlib infeasible linear programs

A comparison of different routing schemes for the robust network loading problem: polyhedral results and computation

International audienceWe consider the capacity formulation of the Robust Network Loading Problem. The aim of the paper is to study what happens from the theoretical and from the computational point of view when the routing policy (or scheme) changes. The theoretical results consider static, volume, affine and dynamic routing, along with splittable and unsplittable flows. Our polyhedral study provides evidence that some well-known valid inequalities (the robust cutset inequalities) are facets for all the considered routing/flows policies under the same assumptions. We also introduce a new class of valid inequalities, the robust 3-partition inequalities, showing that, instead, they are facets in some settings, but not in others. A branch-and-cut algorithm is also proposed and tested. The computational experiments refer to the problem with splittable flows and the budgeted uncertainty set. We report results on several instances coming from real-life networks, also including historical traffic data, as well as on randomly generated instances. Our results show that the problem with static and volume routing can be solved quite efficiently in practice and that, in many cases, volume routing is cheaper than static routing, thus possibly representing the best compromise between cost and computing time. Moreover, unlikely from what one may expect, the problem with dynamic routing is easier to solve than the one with affine routing, which is hardly tractable, even using decomposition methods

Optimizing the investments in mobile networks and subscriber migrations for a telecommunication operator

We consider the context of a telecommunications company that is at the same time an infrastructure operator and a service provider. When planning its network expansion, the company can leverage over its knowledge of subscribers dynamic to better optimize the network dimensioning, therefore avoiding unnecessary costs. In this work, the network expansion represents the deployment and/or reinforcement of several technologies (e.g. 2G,3G,4G), assuming that subscribers to a given technology can be served by this technology or older ones. The operator can influence subscribers dynamic by subsidies. The planning is made over a discretized time horizon while some strategic guidelines requirements are demanded at the end of the time horizon. Following classical models, we consider that the behavior of customers follows an S-shape piecewise constant function. We propose a Mixed-Integer Linear Programming formulation and a heuristic algorithm for the multi-year planning problem. The scalability of the formulation and the quality of the heuristic are assessed numerically on real instances for a use-case with two generations

Min-Max-Min Robustness for Combinatorial Problems with Discrete Budgeted Uncertainty

We consider robust combinatorial optimization problems with cost uncertainty where the decision maker can prepare K solutions beforehand and chooses the best of them once the true cost is revealed. Also known as min-max-min robustness (a special case of K-adaptability), it is a viable alternative to otherwise intractable two-stage problems. The uncertainty set assumed in this paper considers that in any scenario, at most Γ of the components of the cost vectors will be higher than expected, which corresponds to the extreme points of the budgeted uncertainty set. While the classical min-max problem with budgeted uncertainty is essentially as easy as the underlying deterministic problem, it turns out that the min-max-min problem is N P-hard for many easy combinatorial optimization problems, and not approximable in general. We thus present an integer programming formulation for solving the problem through a row-and-column generation algorithm. While exact, this algorithm can only cope with small problems, so we present two additional heuristics leveraging the structure of budgeted uncertainty. We compare our row-and-column generation algorithm and our heuristics on knapsack and shortest path instances previously used in the scientific literature and find that the heuristics obtain good quality solutions in short computational times

Solving the bifurcated and nonbifurcated robust network loading problem with k-adaptive routing

International audienceWe experiment with an alternative routing scheme for the robust network loading problem with demand uncertainty. Named k‐adaptive, it is based on the fact that the decision‐maker chooses k second‐stage solutions and then commits to one of them only after realization of the uncertainty. This routing scheme, with its corresponding k‐partition of the uncertainty set, is dynamically defined under an iterative method to sequentially improve the solution. The method has an inherent characteristic of multiplying the number of variables and constraints after each iteration, so that additional measures are introduced in the solution strategy in order to control time performance. We compare our k‐adaptive results with the ones obtained through other routing schemes and also verify the effectiveness of the methods utilized using several realistic networks from SNDlib and other sources

The resource constrained shortest path problem with uncertain data: a robust formulation and optimal solution approach

International audienceThe Resource Constrained Shortest Path Problem (RCSP P) models several applications in the fields of transportation and communications. The classical problem supposes that the resource consumptions and the costs are certain and looks for the cheapest feasible path. These parameters are however hardly known with precision in real applications, so that the deterministic solution is likely to be infeasible or suboptimal. We address this issue by considering a robust counterpart of the RCSP P. We focus here on resource variation and model its variability through the uncertainty set defined by Bertismas and Sim (2003,2004), which can model the risk aversion of the decision maker through a budget of uncertainty. We solve the resulting problem to optimality through the well-known three phase approach dealing with bounds computation, network reduction and gap closing. In particular, we compute robust bounds on the resource consumption and cost by solving the robust shortest path problem and the dual robust Lagrangian relaxation, respectively. Dynamic programming is used to close the duality gap. Upper and lower bounds are used to reduce the dimension of the network and incorporated in the dynamic programming in order to fathom unpromising states. An extensive computational phase is carried out in order to asses the behavior of the defined strategy comparing its performance with the state-of-the-art. The results highlight the effectiveness of our approach in solving to optimality * 1 benchmark instances for RCSP P when Γ is not too large, tailored for the robust counterpart. For larger values of Γ, we show that the most efficient method combines deterministic preprocesing with the iterative algorithm from Bertsimas and Sim (2003). We also illustrate the failure probability of the robust solutions through Monte Carlo sampling

Minimizing Energy and Link Utilization in ISP Backbone Networks with multi-path Routing: A Bi-level Approach

International audienceIn recent years, green networking has attracted a lot of attention from device manufacturers and Internet Service Providers (ISP) to reduce energy consumption. In the literature, energy-aware traffic engineering problem is proposed to minimize the total energy consumption by switching off unused network devices (routers and links) while guaranteeing full network connectiv-ity. In this work, we are interested in the problem of energy-aware Traffic Engineering while using multi-path routing (ETE-MPR) to minimize link capacity utilization in ISP backbone networks. To this end, we propose a bi-level optimization model where the upper level represents the energy management function , and the lower level refers to the deployed multi-path routing protocol. Then, we reformulate it as a one-level MILP replacing the second level problem by different sets of optimality conditions. We further use these formulations to solve the problem with classical branch-and-bound, cutting plane, and branch-and-cut algorithms. The computational experiments are performed on real instances to compare the proposed algorithms and to evaluate the efficiency of our model against the existing single-path and multi-objective approaches