Network Migration Problem: A Logic-based Benders Decomposition Approach Driven by Column Generation and Constraint Programming

Telecommunication networks frequently face technological advancements and need to upgrade their infrastructure. Adapting legacy networks to the latest technology requires synchronized technicians responsible for migrating the equipment. The goal of the network migration problem is to find an optimal plan for this process. This is a defining step in the customer acquisition of telecommunications service suppliers, and its outcome directly impacts the network owners' purchasing behaviour. We propose the first exact method for the network migration problem, a logic-based Benders decomposition approach that benefits from a hybrid constraint programming-based column generation in its master problem and a constraint programming model in its subproblem. This integrated solution technique is applicable to any integer programming problem with similar structure, most notably the vehicle routing problem with node synchronization constraints. Comprehensive evaluation of our method over instances based on six real networks demonstrates the computational efficiency of the algorithm in obtaining quality solutions. We also show the merit of each incorporated optimization paradigm in achieving this performance

Efficient Spectrum Utilization in Large-Scale RWA and RSA Problems

While the Routing and Wavelength Assignment (RWA) problem has been widely studied, very few studies attempt to solve realistic size instances, namely, with 100 wavelengths per fiber and a few hundred nodes. Indeed, state of the art is closer to around 20 nodes and 30 wavelengths. In this study, we are interested in reducing the gap between realistic data sets and testbed instances, using exact methods. We propose different algorithms that lead to solve exactly or near exactly much larger instances than in the literature, with up to 150 wavelengths and 90 nodes. Extensive numerical experiences are conducted on both the static and the dynamic cases. For the latter, we investigate how much bandwidth is wasted when no lightpath re-arrangement is allowed, and compare it with the number of lightpath re-arrangement it requires in order to fully maximize the grade of service. Results show that the amount of lightpath re-arrangement remains very small in comparison to the amount of wasted bandwidth if not done. The Routing and Spectrum Assignment (RSA) problem is a much more difficult problem than RWA, considered in elastic optical networks. Although investigated extensively, there is still a gap between the size of the instances that can be solved using the current heuristic or exact algorithms, and the size of the instances arising in the industry. As the second objective of this study, we aim to reduce the gap between the two, using a new mathematical modeling, and compare its performance with the best previous algorithms/models on realistic data instances

Lagrangian Dual Decision Rules for Multistage Stochastic Mixed Integer Programming

Multistage stochastic programs can be approximated by restricting policies to follow decision rules. Directly applying this idea to problems with integer decisions is difficult because of the need for decision rules that lead to integral decisions. In this work, we introduce Lagrangian dual decision rules (LDDRs) for multistage stochastic mixed integer programming (MSMIP) which overcome this difficulty by applying decision rules in a Lagrangian dual of the MSMIP. We propose two new bounding techniques based on stagewise (SW) and nonanticipative (NA) Lagrangian duals where the Lagrangian multiplier policies are restricted by LDDRs. We demonstrate how the solutions from these duals can be used to drive primal policies. Our proposal requires fewer assumptions than most existing MSMIP methods. We compare the theoretical strength of the restricted duals and show that the restricted NA dual can provide relaxation bounds at least as good as the ones obtained by the restricted SW dual. In our numerical study, we observe that the proposed LDDR approaches yield significant optimality gap reductions compared to existing general-purpose bounding methods for MSMIP problems

Stochastic RWA and Lightpath Rerouting in WDM Networks

In a telecommunication network, Routing and Wavelength Assignment (RWA) is the problem of finding lightpaths for incoming connection requests. When facing a dynamic traffic, greedy assignment of lightpaths to incoming requests based on predefined deterministic policies leads to a fragmented network that cannot make use of its full capacity due to stranded bandwidth. At this point service providers try to recover the capacity via a defragmentation process. We study this setting from two perspectives: (i) while granting the connection requests via the RWA problem and (ii) during the defragmentation process by lightpath rerouting. For both problems, we present the first two-stage stochastic integer programming model incorporating incoming request uncertainty to maximize the expected grade of service. We develop a decomposition-based solution approach, which uses various relaxations of the problem and a newly developed problem-specific cut family. Simulation of two-stage policies for a variety of instances in a rolling-horizon framework of 52 stages shows that our stochastic models provide high-quality solutions compared to traditionally used deterministic ones. Specifically, the proposed provisioning policies yield improvements of up to 19% in overall grade of service and 20% in spectrum saving, while the stochastic lightpath rerouting policies grant up to 36% more requests using up to just 4% more bandwidth spectrum

Primal and dual decision rules for multi-stage robust optimization

International audienceIn this work, we adapt the recent decision rules introduced in the stochastic programming literature to multi-stage robust optimization both from the primal and the dual perspective. From the primal perspective, we propose two-stage decision rules that restrict the functional forms of state variables only. From the dual perspective, we first write a Lagrangian dual based on the relaxation of non-anticipativity constraints. We then apply decision rules to the Lagrangian multipliers. The resulting problems are challenging and require advanced techniques in their solution. Our methodology is illustrated with preliminary results on production planning and transportation problems

Two-stage and Lagrangian Dual Decision Rules for Multistage Adaptive Robust Optimization

In this work, we design primal and dual bounding methods for multistage adjustable robust optimization (MSARO) problems by adapting two decision rules rooted in the stochastic programming literature. This approach approximates the primal and dual formulations of an MSARO problem with two-stage models. From the primal perspective, this is achieved by applying two-stage decision rules that restrict the functional forms of a certain subset of decision variables. We present sufficient conditions under which the well-known constraint-and-column generation algorithm can be used to solve the primal approximation with finite convergence guarantees. From the dual side, we introduce a distributionally robust dual problem for MSARO models using their nonanticipative Lagrangian dual and then apply linear decision rules on the Lagrangian multipliers. For this dual approximation, we present a monolithic bilinear program valid for continuous recourse problems, and a cutting-plane method for mixed-integer recourse problems. Our framework is general-purpose and does not require strong assumptions such as a stage-wise independent uncertainty set, and can consider integer recourse variables. Computational experiments on newsvendor, location-transportation, and capital budgeting problems show that our bounds yield considerably smaller optimality gaps compared to the existing methods

Two-stage and Lagrangian Dual Decision Rules for Multistage Adaptive Robust Optimization

In this work, we design primal and dual bounding methods for multistage adjustable robust optimization (MSARO) problems by adapting two decision rules rooted in the stochastic programming literature. This approach approximates the primal and dual formulations of an MSARO problem with two-stage models. From the primal perspective, this is achieved by applying two-stage decision rules that restrict the functional forms of a certain subset of decision variables. We present sufficient conditions under which the well-known constraint-and-column generation algorithm can be used to solve the primal approximation with finite convergence guarantees. From the dual side, we introduce a distributionally robust dual problem for MSARO models using their nonanticipative Lagrangian dual and then apply linear decision rules on the Lagrangian multipliers. For this dual approximation, we present a monolithic bilinear program valid for continuous recourse problems, and a cutting-plane method for mixed-integer recourse problems. Our framework is general-purpose and does not require strong assumptions such as a stage-wise independent uncertainty set, and can consider integer recourse variables. Computational experiments on newsvendor, location-transportation, and capital budgeting problems show that our bounds yield considerably smaller optimality gaps compared to the existing methods

Preparation and anatomical distribution study of 67Ga-alginic acid nanoparticles for SPECT purposes in rainbow trout (Oncorhynchus mykiss)

Ergosan contains 1% alginic acid extracted from two brown sea weeds. Little is known about the target organs and anatomical distribution of Ergosan (alginic acid) in fish. Therefore, feasibility of developing alginic acid nanoparticles to detect target organ in rainbow trout is interesting. To make nanoparticles, Ergosan extract (alginic acid) was irradiated at 30 kGy in a cobalt-60 irradiator and characterized by transmission electron microscopy (TEM) and Fourier transform infrared spectroscopy (FTIR). Results from TEM images showed that particle sizes of irradiated alginic acid ranged from 30 to 70 nm. The FTIR results indicated that gamma irradiation had no significant influence on the basic structure of alginic acid. Later, alginic acid nanoparticles were successively labelled with 67Ga-gallium chloride. The biodistribution of irradiated Ergosan in normal rainbow trout showed highest uptake in intestine and kidney and then in liver and kidney at 4- and 24-h post injection, respectively. Single-photon emission computed tomography (SPECT) images also demonstrated target specific binding of the tracer at 4- and 24-h post injection. In conclusion, the feed supplemented with alginic acid nanoparticles enhanced SPECT images of gastrointestinal morphology and immunity system in normal rainbow trout