Publications

2023

• Relaxed-inertial proximal point type algorithms for problems involving strongly quasiconvex functions
  
  • Grad Sorin-Mihai
  • Lara Felipe
  • Marcavillaca Raul Tintaya
  , 2023. Introduced in the 1970's by Martinet for minimizing convex functions and extended shortly afterwards by Rockafellar towards monotone inclusion problems, the proximal point algorithm turned out to be a viable computational method for solving various classes of optimization problems even beyond the convex framework. In this talk we propose a relaxed-inertial proximal point type algorithm for solving optimization problems consisting in minimizing strongly quasiconvex functions whose variables lie in finitely dimensional linear subspaces. The method is then extended for equilibrium functions involving strongly quasiconvex functions. Computational results confirm the theoretical advances.
• Diffraction électromagnétique par une couche mince de nanoparticules réparties aléatoirement : développement asymptotique, conditions effectives et simulations.
  
  • Boucart Amandine
  , 2023. Nous considérons le problème de diffraction, en régime harmonique, d’une onde plane électromagnétique par un objet inhomogène recouvert d’une couche très fine de petites particules parfaitement conductrices distribuées aléatoirement. Nous cherchons à quantifier l’effet de cette couche sur le coefficient de réflexion. La taille des particules, leur espacement et l’épaisseur de la couche sont du même ordre mais petites par rapport à la longueur d’onde incidente et les dimensions de l’objet. Deux difficultés apparaissent : (1) Résoudre numériquement les équations de Maxwell dans ce contexte est extrêmement coûteux en terme de taille mémoire et de temps calcul; (2) la répartition des particules n'est pas connue pour un objet donné. Nous allons supposer que c'est une réalisation d'une répartition supposée aléatoire.Pour contourner ces difficultés, nous proposons alors un modèle effectif, à l’aide d’un développement asymptotique multi-échelle de la solution, où la couche de particules est remplacée par une condition aux bords effective, prescrite sur une surface située au-dessus des particules. Les coefficients qui interviennent dans la condition nécessite la résolution de problèmes, dits de cellule, posés un demi-espace recouvert d'une couche de particules, de taille unitaire, réparties aléatoirement.
• Scattering resonances in unbounded transmission problems with sign-changing coefficient
  
  • Carvalho Camille
  • Moitier Zoïs
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2023, 88 (2), pp.215-257. It is well-known that classical optical cavities can exhibit localized phenomena associated to scattering resonances, leading to numerical instabilities in approximating the solution. This result can be established via the ``quasimodes to resonances'' argument from the black-box scattering framework. Those localized phenomena concentrate at the inner boundary of the cavity and are called whispering gallery modes. In this paper we investigate scattering resonances for unbounded transmission problems with sign-changing coefficient (corresponding to optical cavities with negative optical properties, for example made of metamaterials). Due to the change of sign of optical properties, previous results cannot be applied directly, and interface phenomena at the metamaterial-dielectric interface (such as the so-called surface plasmons) emerge. We establish the existence of scattering resonances for arbitrary two-dimensional smooth metamaterial cavities. The proof relies on an asymptotic characterization of the resonances, and showing that problems with sign-changing coefficient naturally fit the black box scattering framework. Our asymptotic analysis reveals that, depending on the metamaterial's properties, scattering resonances situated closed to the real axis are associated to surface plasmons. Examples for several metamaterial cavities are provided. (10.1093/imamat/hxad005)
  DOI : 10.1093/imamat/hxad005
• Relaxed-inertial proximal point algorithms for problems involving strongly quasiconvex functions
  
  • Grad Sorin-Mihai
  • Lara Felipe
  • Marcavillaca Raul Tintaya
  , 2023.
• Analysis of sampling methods for imaging a periodic layer and its defects
  
  • Boukari Yosra
  • Haddar Houssem
  • Jenhani Nouha
  Inverse Problems, IOP Publishing, 2023, 39 (5), pp.055001. We revisit the differential sampling method introduced in [9] for the identification of a periodic domain and some local perturbation. We provide a theoretical justification of the method that avoids assuming that the local perturbation is also periodic. Our theoretical framework uses functional spaces with continuous dependence with respect to the Floquet-Bloch variable. The corner stone of the analysis is the justification of the Generalized Linear Sampling Method in this setting for a single Floquet-Bloch mode. (10.1088/1361-6420/acc19a)
  DOI : 10.1088/1361-6420/acc19a
• Lecture notes on numerical linear algebra
  
  • Bonnet Marc
  , 2023. Course notes, ENSTA Paris (2nd year and M1 level), 2021.
• Fourier representation of the diffusion MRI signal using layer potentials
  
  • Fang Chengran
  • Wassermann Demian
  • Li Jing-Rebecca
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2023, 83 (1), pp.99-121. The diffusion magnetic resonance imaging signal arising from biological tissues can be numerically simulated by solving the Bloch-Torrey partial differential equation. Numerical simulations can facilitate the investigation of the relationship between the diffusion MRI signals and cellular structures. With the rapid advance of available computing power, the diffusion MRI community has begun to employ numerical simulations for model formulation and validation, as well as for imaging sequence optimization. Existing simulation frameworks use the finite difference method, the finite element method, or the Matrix Formalism method to solve the Bloch-Torrey partial differential equation. We propose a new method based on the efficient evaluation of layer potentials. In this paper, the mathematical framework and the numerical implementation of the new method are described. We demonstrate the convergence of our method via numerical experiments and analyze the errors linked to various model and simulation parameters. Since our method provides a Fourier-type representation of the diffusion MRI signal, it can potentially facilitate new physical and biological signal interpretations in the future. (10.1137/21M1439572)
  DOI : 10.1137/21M1439572
• Operations Research Approaches for the Satellite Constellation Design Problem
  
  • Mencarelli Luca
  • Floquet Julien
  • Georges Frédéric
  , 2023.
• Neuron modeling, Bloch-Torrey equation, and their application to brain microstructure estimation using diffusion MRI
  
  • Fang Chengran
  , 2023. Non-invasively estimating brain microstructure that consists of a very large number of neurites, somas, and glial cells is essential for future neuroimaging. Diffusion MRI (dMRI) is a promising technique to probe brain microstructural properties below the spatial resolution of MRI scanners. Due to the structural complexity of brain tissue and the intricate diffusion MRI mechanism, in vivo microstructure estimation is challenging.Existing methods typically use simplified geometries, particularly spheres, and sticks, to model neuronal structures and to obtain analytical expressions of intracellular signals. The validity of the assumptions made by these methods remains undetermined. This thesis aims to facilitate simulationdriven brain microstructure estimation by replacing simplified geometries with realistic neuron geometry models and the analytical intracellular signal expressions with diffusion MRI simulations. Combined with accurate neuron geometry models, numerical dMRI simulations can give accurate intracellular signals and seamlessly incorporate effects arising from, for instance, neurite undulation or water exchange between soma and neurites.Despite these advantages, dMRI simulations have not been widely adopted due to the difficulties in constructing realistic numerical phantoms, the high computational cost of dMRI simulations, and the difficulty in approximating the implicit mappings between dMRI signals and microstructure properties. This thesis addresses the above problems by making four contributions. First, we develop a high-performance opensource neuron mesh generator and make publicly available over a thousand realistic cellular meshes.The neuron mesh generator, swc2mesh, can automatically and robustly convert valuable neuron tracing data into realistic neuron meshes. We have carefully designed the generator to maintain a good balance between mesh quality and size. A neuron mesh database, NeuronSet, which contains 1213 simulation-ready cell meshes and their neuroanatomical measurements, was built using the mesh generator. These meshes served as the basis for further research. Second, we increased the computational efficiency of the numerical matrix formalism method by accelerating the eigendecomposition algorithm and exploiting GPU computing. The speed was increased tenfold. With similar accuracy, the optimized numerical matrix formalism is 20 times faster than the FEM method and 65 times faster than a GPU-based Monte Carlo method. By performing simulations on realistic neuron meshes, we investigated the effect of water exchange between somas and neurites, and the relationship between soma size and signals. We then implemented a new simulation method that provides a Fourier-like representation of the dMRI signals. This method was derived theoretically and implemented numerically. We validated the convergence of the method and showed that the error behavior is consistent with our error analysis. Finally, we propose a simulation-driven supervised learning framework to estimate brain microstructure using diffusion MRI. By exploiting the powerful modeling and computational capabilities that are mentioned above, we have built a synthetic database containing the dMRI signals and microstructure parameters of 1.4 million artificial brain voxels. We have shown that this database can help approximate the underlying mappings of the dMRI signals to volume and surface fractions using artificial neural networks.
• Scattering in a partially open waveguide: the forward problem
  
  • Bourgeois Laurent
  • Fliss Sonia
  • Fritsch Jean-François
  • Hazard Christophe
  • Recoquillay Arnaud
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2023, 88, pp.102-151. This paper is dedicated to an acoustic scattering problem in a two-dimensional partially open waveguide, in the sense that the left part of the waveguide is closed, that is with a bounded cross-section, while the right part is bounded in the transverse direction by some Perfectly Matched Layers that mimic the situation of an open waveguide, that is with an unbounded cross-section. We prove well-posedness of such scattering problem and exhibit the asymptotic behaviour of the solution in the longitudinal direction with the help of the Kondratiev approach. Having in mind the numerical computation of the solution, we also propose some transparent boundary conditions in such longitudinal direction, based on Dirichlet-to-Neumann operators. After proving that such artificial conditions actually enable us to approximate the exact solution, some numerical experiments illustrate the quality of such approximation. (10.1093/imamat/hxad004)
  DOI : 10.1093/imamat/hxad004
• Inside-Outside Duality for Modified Transmission Eigenvalues
  
  • Haddar Houssem
  • Khenissi Moez
  • Mansouri Marwa
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2023, 17 (4), pp.798-816. We introduce a new modified spectrum associated with the scattering from penetrable objects using the modified background technique. We prove that the inside-outside duality method allows to reconstruct this spectrum from full aperture far field measurements at a fixed frequency. The method is numerically tested and validated on some synthetic examples. (10.3934/ipi.2023004)
  DOI : 10.3934/ipi.2023004
• Analysis of time-harmonic Maxwell impedance problems in anisotropic media
  
  • Chicaud Damien
  • Ciarlet Patrick
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2023, 55 (3), pp.1969-2000. We consider the time-harmonic Maxwell's equations in anisotropic media. The problem to be solved is an approximation of the diffraction problem, or scattering from bounded objects, that is usually set in some exterior domain in $\mathbb{R}^3$. We consider perfectly conducting objects, so the equations are supplemented with a Dirichlet boundary condition on those objects, and we truncate the exterior domain by imposing an impedance condition on an artificial boundary, to model an approximate radiation condition. The resulting problem is then posed in a bounded domain, with Dirichlet and impedance boundary conditions. In this work, we focus on the mathematical meaning of the impedance condition, precisely in which function space it holds. This relies on a careful analysis of the regularity of the traces of electromagnetic fields, which can be derived thanks to the study of the regularity of the solution to second-order surface PDEs. Then, we prove well-posedness of the model, and we determine the a priori regularity of the fields in the domain and on the boundaries, depending on the geometry, the coefficients and the data. Finally, the discretization of the formulations is presented, with an approximation based on edge finite elements. Error estimates are derived, and a benchmark is provided to discuss those estimates. (10.1137/22M1485413)
  DOI : 10.1137/22M1485413
• Asymptotic models of the diffusion MRI signal accounting for geometrical deformations
  
  • Yang Zheyi
  • Mekkaoui Imen
  • Hesthaven Jan
  • Li Jing-Rebecca
  MathematicS In Action, Société de Mathématiques Appliquées et Industrielles (SMAI), 2023, 12 (1), pp.65-85. The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses can be modeled by the Bloch-Torrey partial differential equation (PDE). The associated diffusion MRI signal is the spatial integral of the solution of the Bloch-Torrey PDE. In addition to the signal, the time-dependent apparent diffusion coefficient (ADC) can be obtained from the solution of another partial differential equation, called the HADC model, which was obtained using homogenization techniques. In this paper, we analyze the Bloch-Torrey PDE and the HADC model in the context of geometrical deformations starting from a canonical configuration. To be more concrete, we focused on two analytically defined deformations: bending and twisting. We derived asymptotic models of the diffusion MRI signal and the ADC where the asymptotic parameter indicates the extent of the geometrical deformation. We compute numerically the first three terms of the asymptotic models and illustrate the effects of the deformations by comparing the diffusion MRI signal and the ADC from the canonical configuration with those of the deformed configuration. The purpose of this work is to relate the diffusion MRI signal more directly with tissue geometrical parameters. (10.5802/msia.32)
  DOI : 10.5802/msia.32
• Wave propagation in one-dimensional quasiperiodic media
  
  • Amenoagbadji Pierre
  • Fliss Sonia
  • Joly Patrick
  Communications in Optimization Theory, Mathematical Research Press, 2023, 17. This work is devoted to the resolution of the Helmholtz equation −(µ u) − ρ ω 2 u = f in a one-dimensional unbounded medium. We assume the coefficients of this equation to be local perturbations of quasiperiodic functions, namely the traces along a particular line of higher-dimensional periodic functions. Using the definition of quasiperiodicity, the problem is lifted onto a higher-dimensional problem with periodic coefficients. The periodicity of the augmented problem allows us to extend the ideas of the DtN-based method developed in [10, 19] for the elliptic case. However, the associated mathematical and numerical analysis of the method are more delicate because the augmented PDE is degenerate, in the sense that the principal part of its operator is no longer elliptic. We also study the numerical resolution of this PDE, which relies on the resolution of Dirichlet cell problems as well as a constrained Riccati equation. (10.23952/cot.2023.17)
  DOI : 10.23952/cot.2023.17
• A General Comparison Principle for Hamilton Jacobi Bellman Equations on Stratified Domains
  
  • Jerhaoui Othmane
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2023, 29. This manuscript aims to study finite horizon, first order Hamilton Jacobi Bellman (HJB) equations on stratified domains. This problem is related to optimal control problems with discontinuous dynamics. We use nonsmooth analysis techniques to derive a strong comparison principle as in the classical theory and deduce that the value function is the unique viscosity solution. Furthermore, we prove some stability results of the Hamilton Jacobi Bellman equation. Finally, we establish a general convergence result for monotone numerical schemes in the stratified case. (10.1051/cocv/2022089)
  DOI : 10.1051/cocv/2022089
• An optimal control-based numerical method for scalar transmission problems with sign-changing coefficients
  
  • Ciarlet Patrick
  • Lassounon David
  • Rihani Mahran
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2023, 61 (3), pp.1316-1339. In this work, we present a new numerical method for solving the scalar transmission problem with sign-changing coefficients. In electromagnetism, such a transmission problem can occur if the domain of interest is made of a classical dielectric material and a metal or a metamaterial, with for instance an electric permittivity that is strictly negative in the metal or metamaterial. The method is based on an optimal control reformulation of the problem. Contrary to other existing approaches, the convergence of this method is proved without any restrictive condition. In particular, no condition is imposed on the a priori regularity of the solution to the problem, and no condition is imposed on the meshes, other than that they fit with the interface between the two media. Our results are illustrated by some (2D) numerical experiments. (10.1137/22M1495998)
  DOI : 10.1137/22M1495998
• Time Blocks Decomposition of Multistage Stochastic Optimization Problems
  
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  • Martin Thomas
  • Rigaut Tristan
  Journal of Convex Analysis, Heldermann, 2023, 30 (2), pp.627-658. Multistage stochastic optimization problems are, by essence, complex as their solutions are indexed both by stages and by uncertainties. Their large scale nature makes decomposition methods appealing, like dynamic programming which is a sequential decomposition using a state variable defined at all stages. In this paper, we introduce the notion of state reduction by time blocks, that is, at stages that are not necessarily all the original stages. Then, we prove a reduced dynamic programming equation. We position our result with respect to the most well-known mathematical frameworks for dynamic programming. We illustrate our contribution by showing its potential for applied problems with two time scales.
• McKean SDEs with singular coefficients
  
  • Issoglio Elena
  • Russo Francesco
  Annales de l'Institut Henri Poincaré, Presses universitaires de France — PUF, 2023, 59 (3), pp.1530-1548. The paper investigates existence and uniqueness for a stochastic differential equation (SDE) with distributional drift depending on the law density of the solution. Those equations are known as McKean SDEs. The McKean SDE is interpreted in the sense of a suitable singular martingale problem. A key tool used in the investigation is the study of the corresponding Fokker-Planck equation. (10.1214/22-AIHP1293)
  DOI : 10.1214/22-AIHP1293
• Scattering in a partially open waveguide: the inverse problem
  
  • Bourgeois Laurent
  • Fritsch Jean-François
  • Recoquillay Arnaud
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2023, 17 (2), pp.463-469. In this paper we consider an inverse scattering problem which consists in retrieving obstacles in a partially embedded waveguide in the acoustic case, the measurements being located on the accessible part of the structure. Such accessible part can be considered as a closed waveguide (with a finite cross section), while the embedded part can be considered as an open waveguide (with an infinite cross section). We propose an approximate model of the open waveguide by using Perfectly Matched Layers in order to simplify the resolution of the inverse problem, which is based on a modal formulation of the Linear Sampling Method. Some numerical results show the efficiency of our approach. This paper can be viewed as a continuation of the article [11], which was focused on the forward problem. (10.3934/ipi.2022052)
  DOI : 10.3934/ipi.2022052
• Ultrasonic imaging in highly heterogeneous backgrounds
  
  • Pourahmadian Fatemeh
  • Haddar Houssem
  Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences, Royal Society, The, 2023, 479 (2271). This work formally investigates the differential evolution indicators as a tool for ultrasonic tracking of elastic transformation and fracturing in randomly heterogeneous solids. Within the framework of periodic sensing, it is assumed that the background at time t◦ contains (i) a multiply connected set of viscoelastic, anisotropic, and piece-wise homogeneous inclusions, and (ii) a union of possibly disjoint fractures and pores. The support, material properties, and interfacial condition of scatterers in (i) and (ii) are unknown, while elastic constants of the matrix are provided. The domain undergoes progressive variations of arbitrary chemo-mechanical origins such that its geometric configuration and elastic properties at future times are distinct. At every sensing step t◦, t1, . . ., multi-modal incidents are generated by a set of boundary excitations, and the resulting scattered fields are captured over the observation surface. The test data are then used to construct a sequence of wavefront densities by solving the spectral scattering equation. The incident fields affiliated with distinct pairs of obtained wavefronts are analyzed over the stationary and evolving scatterers for a suit of geometric and elastic evolution scenarios entailing both interfacial and volumetric transformations. The main theorem establishes the invariance of pertinent incident fields at the loci of static fractures and inclusions between a given pair of time steps, while certifying variation of the same fields over the modified regions. These results furnish a basis for theoretical justification of differential evolution indicators for imaging in complex composites which, in turn, enable the exclusive tomography of evolution in a background endowed with many unknown features. (10.1098/rspa.2022.0721)
  DOI : 10.1098/rspa.2022.0721
• Quantifying mixing in arbitrary fluid domains: a Padé approximation approach
  
  • Anderson Thomas G
  • Bonnet Marc
  • Veerapaneni Shravan
  Numerical Algorithms, Springer Verlag, 2023, 93, pp.441-458. We consider the model problem of mixing of passive tracers by an incompressible viscous fluid. Addressing questions of optimal control in realistic geometric settings or alternatively the design of fluid-confining geometries that successfully effect mixing requires a meaningful norm in which to quantify mixing that is also suitable for easy and efficient computation (as is needed, e.g., for use in gradient-based optimization methods). We use the physically inspired reasonable surrogate of a negative index Sobolev norm over the complex fluid mixing domain Ω, a task which could be seen as computationally expensive since it requires the computation of an eigenbasis for L2(Ω) by definition. Instead, we compute a representant of the scalar concentration field in an appropriate Sobolev space in order to obtain an equivalent definition of the Sobolev surrogate norm. The representant, in turn, can be computed to high-order accuracy by a Padé approximation to certain fractional pseudo-differential operators, which naturally leads to a sequence of elliptic problems with an inhomogeneity related to snapshots of the time-varying concentration field. Fast and accurate potential theoretic methods are used to efficiently solve these problems, with rapid per-snapshot mix-norm computation made possible by recent advances in numerical methods for volume potentials. We couple the methodology to existing solvers for Stokes and advection equations to obtain a unified framework for simulating and quantifying mixing in arbitrary fluid domains. We provide numerical results demonstrating the convergence of the new approach as the approximation order is increased. (10.1007/s11075-022-01423-7)
  DOI : 10.1007/s11075-022-01423-7
• Robust capacitated Steiner trees and networks with uniform demands
  
  • Bentz Cédric
  • Costa Marie-Christine
  • Poirion Pierre‐louis
  • Ridremont Thomas
  Networks, Wiley, 2023, 82 (1), pp.3-31. We are interested in the design of robust (or resilient) capacitated rooted Steiner networks in the case of terminals with uniform demands. Formally, we are given a graph, capacity, and cost functions on the edges, a root, a subset of vertices called terminals, and a bound k on the number of possible edge failures. We first study the problem where k=1 and the network that we want to design must be a tree covering the root and the terminals: we give complexity results and propose models to optimize both the cost of the tree and the number of terminals disconnected from the root in the worst case of an edge failure, while respecting the capacity constraints on the edges. Secondly, we consider the problem of computing a minimum-cost survivable network, that is, a network that covers the root and terminals even after the removal of any k edges, while still respecting the capacity constraints on the edges. We also consider the possibility of protecting a given number of edges. We propose three different formulations: a bilevel formulation (with an attacker and a defender), a cutset-based formulation and a flow-based one. We compare the formulations from a theoretical point of view, and we propose algorithms to solve them and compare their efficiency in practice. (10.1002/net.22143)
  DOI : 10.1002/net.22143
• EMSx: a numerical benchmark for energy management systems
  
  • Le Franc Adrien
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  Energy Systems, Springer, 2023, 14, pp.817–843. Inserting renewable energy in the electric grid in a decentralized manner is a key challenge of the energy transition. However, at local scale, both production and demand display erratic behavior, which makes it delicate to match them. It is the goal of Energy Management Systems (EMS) to achieve such balance at least cost. We present EMSx, a numerical benchmark for testing control algorithms for the management of electric microgrids equipped with a photovoltaic unit and an energy storage system. EMSx is made of three key components: the EMSx dataset, provided by Schneider Electric, contains a diverse pool of realistic microgrids with a rich collection of historical observations and forecasts; the EMSx mathematical framework is an explicit description of the assessment of electric microgrid control techniques and algorithms; the EMSx software EMSx.jl is a package, implemented in the Julia language, which enables to easily implement a microgrid controller and to test it. All components of the benchmark are publicly available, so that other researchers willing to test controllers on EMSx may reproduce experiments easily. Eventually, we showcase the results of standard microgrid control methods, including Model Predictive Control, Open Loop Feedback Control and Stochastic Dynamic Programming. (10.1007/s12667-020-00417-5)
  DOI : 10.1007/s12667-020-00417-5
• Time-Varying Optimization of Networked Systems With Human Preferences
  
  • Ospina Ana
  • Simonetto Andrea
  • Dall'Anese Emiliano
  IEEE Transactions on Control of Network Systems, IEEE, 2023, 10 (1), pp.503 - 515. (10.1109/TCNS.2022.3203467)
  DOI : 10.1109/TCNS.2022.3203467
• Robust MILP formulations for the two-stage weighted vertex p-center problem
  
  • Duran-Mateluna Cristian
  • Ales Zacharie
  • Elloumi Sourour
  • Jorquera-Bravo Natalia
  Computers and Operations Research, Elsevier, 2023, pp.106334. The weighted vertex p-center problem (PCP) consists of locating p facilities among a set of potential sites such that the maximum weighted distance from any client to its closest open facility is minimized. This paper studies the exact resolution of the two-stage robust weighted vertex p-center problem (RPCP2). In this problem, the opening of the centers is fixed in the first stage while the client allocations are recourse decisions fixed once the uncertainty is revealed. The problem uncertainty comes from both the nodal demands and the edge lengths. It is modeled by box uncertainty sets. We introduce three different robust reformulations based on MILPs from the literature. We prove that considering a finite subset of scenarios is sufficient to obtain an optimal solution of (RPCP2). We leverage this result to introduce a column-and-constraint generation algorithm and a branch-and-cut algorithm to efficiently solve this problem optimally. We highlight how these algorithms can be adapted to solve, for the first time to optimality, the single-stage problem (RPCP1) which is obtained when no recourse is considered. We present a numerical study to compare the performance of these formulations on randomly generated instances and a case study from the literature. (10.1016/j.cor.2023.106334)
  DOI : 10.1016/j.cor.2023.106334