3.1 Decomposition methods

Dantzig-Wolfe decomposition 36 is a well-established approach for solving hard and/or large-scale combinatorial optimization problems. Problems that are originally formulated as Mixed-Integer Programming problems (MIPs) are reformulated by defining a (generally exponential) number of variables which correspond to the solutions of subproblem(s) obtained by subsets of variables and constraints of the original MIP formulation. To solve the linear approximation of the Dantzig-Wolfe reformulation, the column generation procedure is employed, which iteratively generates missing variables by solving the subproblem(s). Cut generation is then employed to strengthen the linear approximation, and branching is performed to find an optimal solution to the problem. The overall approach is called the branch-cut-and-price (BCP) method 39. Recent advances, including those developed by our previous project team ReAlOpt, helped significantly improve the efficiency of general-purpose BCP algorithms. To use these algorithms one just has to specify the master problem, and a formulation or an algorithm to solve the column generation subproblem. The improvements include stabilization 62, pre-processing, diving heuristics 70, and strong branching, among others. Generic BCP codes implementing these improvements made the branch-cut-and-price approach more accessible to users.

A major limitation of BCP algorithms is their inefficiency when they face constraints that break the decomposable structure of the problem. Prominent examples are synchronisation and precedence constraints 40. Moreover, it became clear recently that many efficient BCP approaches are "non-robust", i.e., column and cut generation are interdependent. Thus, for every subproblem type, dedicated cut-generation approaches should be considered. Few such approaches are known in the literature 37,60. Their development for different families of cutting planes and different subproblem types is a very promising research direction. Devising BCP algorithms for problems that do not have the decomposability property is a difficult and high-risk task. However, in the case of success, the impact on the practice of Operations Research would be significant.

Our numerical experience shows that generic BCP approaches 45,46 (in which the subproblems are solved as generic MIPs) are rarely competitive with the traditional branch-and-cut method used in widely available and highly efficient MIP solvers. On the other hand, very efficient BCP approaches specialized for particular problems exist in the scientific literature. In such approaches, subproblems are of a certain type and are solved by fast specialized algorithms. These ad-hoc BCP approaches are however rarely used in practice due to the complexity of their implementation 60, 68. An important goal of our project is to propose innovative methods that have the efficiency of the most advanced specialized algorithms in the literature, yet can be applied to many practical cases. This can be done by developing BCP algorithms which are not completely generic (i.e., they cannot be applied to any model obtained by Dantzig-Wolfe reformulation) but can be used for large classes of combinatorial optimization problems. There is a clear need for efficient and reusable semi-generic approaches, that can be applied to a large number of problems inside a specific class. We have identified several types of subproblems that are often encountered in practical applications as substructures: resource-constrained paths in graphs 51 and hyper-graphs 34, constrained network flows, and decision diagrams 25. Developing efficient algorithms that can be applied to the broadest possible class of such subproblems is a hard task. We will seek the right trade-off between generality and efficiency.

Recently, we have developed the first "semi-generic" BCP solver, VRPSolver, which relies on solving resource constrained shortest path subproblems 61. We have shown that this solver can be successfully applied to many vehicle routing variants and some packing problems. An open question is whether there are other classes of problems which can be efficiently solved by "semi-specialized" BCP solvers. Scheduling and network design problems are good candidates for such classes. Capabilities of current BCP solvers however need to be extended to efficiently handle these problems. One of the drawbacks of our solver is the absence of good embedded heuristics to quickly find feasible solutions. Diving heuristics 70 can in principle be used, but their speed and performance are generally not satisfactory.

Another challenge is to design a modelling paradigm that can produce formulations for "semi-specialized" BCP solvers. In this context, defining a MIP and applying the standard Danzig-Wolfe reformulation technique is cumbersome as the subproblems cannot be easily defined as MIPs. New innovative modelling paradigms should be developed to simplify the problem definition and thus extend the usage of BCP approaches. They should be simple enough to attract non-specialists in the domain, and complex enough to pass the structural information to the BCP solver. This is mandatory to achieve state-of-the-art performance. One can find inspiration in the notion of global constraints designed in the field of Constraint Programming.

Benders' decomposition 24 is also becoming increasingly effective for several benchmark problems. Recent methods based on this reformulation approach involve solving many similar subproblems repeatedly, and make use of sophisticated stabilization techniques. We plan to build on the knowledge developed on Dantzig-Wolfe decomposition to develop techniques to improve Benders' decomposition, which do not use any assumption on the specific structure of the subproblem. A considerable challenge is to find effective methods for handling integer subproblems in a generic manner.

Another way to improve Benders' decomposition tools is to use machine learning techniques. Such algorithms can also be used to parameterize the different parts of the method or discover some information to help the convergence of these algorithms. This is a promising path for solving stochastic programs with a large number of scenarios. Several scientific questions have to be addressed, including the possible adaptation of stabilization techniques that need a complete dual information for proving the convergence.

In line with the challenges highlighted above, one of our first objectives is to improve the genericity of our BCP solver for the resource constrained path structure. For the moment, it supports only models with standard additive resources. We intend also to cover the case when cost and resource consumption of the current arc in the path depends not only on the arc itself, but also on the accumulated resource consumption of previous arcs in the path. This would allow to solve several important classes of problems such as electric vehicle routing problems, and scheduling problems with additive objective functions much more efficiently. This implies improving both the algorithms used and the modelling capabilities of the solver to support more general resources.

In the longer term, we plan to develop semi-generic BCP approaches and solvers for problems which do not contain the resource constrained path structure in graphs. Alternative structures which can be considered are spanning trees, network-flow problems in hyper-graphs, context-free grammars (both mentioned in Section 3.2), and decision diagrams.

Another interesting idea to explore in this direction is developing decomposition approaches for multi-stage sequential decision-making problems under uncertainty. Such problems can be modeled as multi-agent Markov Decision Processes (MDP). We believe that mathematical programming decomposition algorithms in which sub-problems are single-agent MDPs solved by dynamic programming constitute a very promising alternative to currently used approaches for multi-agent MDPs. Moreover, there are currently no approaches for such problems which provide exact solutions and non-trivial valid bounds on the optimal value.

3.2 Extended formulations

Many successful extended formulations are based on network-flow models. These models have excellent polyhedral properties, since the constraint matrix of a flow problem is totally unimodular, and thus all extreme-point solutions of the linear program related to this subsystem are integer if the right-hand sides of the constraints are integer. Additional constraints generally break this property, but in many cases, the linear relaxation remains of excellent quality.

Network-flow reformulations can be obtained when all or parts of discrete optimization problems can be described by regular or algebraic languages, or by recursive (such as dynamic programming) formulations, on top of which additional constraints such as resource constraints, disjunctions, are specified. Karp and Held 52 provided a systematic approach to building dynamic programming recurrence equations for a large class of optimization problems by characterizing the representation of discrete decision processes by monotone sequential decision processes. Martin et al. 58 characterized discrete optimization problems that can be solved via dynamic programming using directed hypergraphs 21. The formalism of sequential decision processes allows to build an internal representation of the sequential structure using network-flows in state-graphs or hypergraphs. Additional constraints can be embedded in this representation by increasing the size of the network, for example by discretizing quantities such as time (scheduling), load (vehicle routing), or width (bin-packing) 47. This has several advantages: 1) enforcing the consideration of constraints using the graph structure (that is convexifying these constraints), 2) easing the modeling of complex constraints, 3) facilitating the consideration of resource-dependent parameters 72.

The main drawback of these formulations is their typically exponential or pseudo-polynomial number of variables and constraints. This is different from models that arise from a Dantzig-Wolfe decomposition, which have an exponential number of variables, but a generally small number of constraints. The size of network-flow models is usually far too large to make them solvable by modern integer programming, SAT or constraint programming solvers.

To overcome this limitation, aggregation and disaggregation techniques, based either on a scaling of the input parameters or on aggregating nodes of the network, are good candidates. As they generally make use of sub-routines, they allow an efficient hybridization of different optimization paradigms. Several methods sharing common ideas have been introduced in different communities. For solving dynamic programs, techniques based on iterative state-space relaxations 32 have proved their efficiency since the 1980s. The most popular methods are the decremental state-space relaxation 28, 67 and the successive sublimation dynamic programming 50. In the field of MIP models, the most promising algorithms for solving exponentially-large network-flow formulations are also based on an increasingly refined series of relaxations (see 33,29,66). These methods produce initial models with fewer variables and fewer constraints than the original ones and iteratively refine them after identifying new variables and/or constraints to add. Although these methods have been shown to be efficient for several benchmark problems, only few generic methods have been proposed (see 30 and 54).

Iterative aggregation/disaggregation techniques have some relationship to other methods. They share similarities with Benders' decomposition or Logic-based Benders' decomposition. Both methods rely initially on a relaxation of the problem. However, in Benders' decomposition constraints are iteratively introduced to discard infeasible solutions, while iterative aggregation/disaggregation-based techniques may also introduce new variables. Similarities with optimization methods using decision diagrams constructed by (iterative) state aggregation procedures can also be highlighted 25. Decision diagrams are graphs storing possible variable assignments satisfying some constraints and are particularly used in CP/SAT solvers.

A conclusion of the above is that similar aggregation and disaggregation techniques are used under different names in different fields (MIP, DP, CP/SAT), leading to a scattered scientific literature. Further, most of the proposed techniques are problem-dependent (with the notable exception of decision diagrams, which offer a more general setting, although we are not aware of any generic implementation). Our goal is to go in the direction of developing a unifying formalism and express the main methods of the literature within this formalism. This generic view aims to bring together methods whose proximity has never really been highlighted. Existing algorithms may benefit from algorithmic components that have proven their effectiveness in other contexts. As an example, existing MIP algorithms could benefit from state-of-the-art approaches developed for decision diagrams.

We will study and design effective methods based on aggregating and disaggregating models. It is necessary to develop a high-level modeling tool to specify these models only implicitly. The use of algebraic languages to produce such higher-level formulations is a direction we are considering to explore. We also believe that a primal-dual solution approach is worth exploring. Such an approach will rely on the local refinement of a coarse representation of the global problem when needed. The idea is to derive primal and dual bounds on the objective value of the problem with a lower computational effort compared to working with the original model. We can distinguish two types of aggregation: 1) a conservative aggregation ensures that every solution of the original model is also a solution of the aggregated model (i.e., the aggregated model is a relaxation of the original model) and 2) a heuristic aggregation ensures that every solution of the aggregated model is also a solution of the original model. After projecting the original model onto an aggregated one, we aim to converge to an optimal solution without reaching the size of the original model. To keep the model tractable, one can make use of primal and dual information from the different aggregated models to fix variables values (using Lagrangian filtering for example). We will focus on finding a systematic way to 1) aggregate an initial model to yield either a relaxation or a restriction and 2) iteratively disaggregate the current model for obtaining better dual bounds. A path worth exploring is to dynamically aggregate the models taking advantage of the information learned in the current phase to build stronger models 49. We will also study an emerging technique consisting in the joint use of several aggregated models with decision synchronization 59,55 which can be helpful to derive stronger bounds and to keep tractable network-flow models.

For designing heuristics, the problems generated by dynamic programs / algebraic languages are suited to so-called structured learning techniques, where the objective is to learn how to approximate a hard combinatorial problem (typically a sequencing problem with resource constraints) by means of a simpler problem (typically a shortest-path problem on a graph). More classical learning techniques can also be used to guess the right parameters (lagrangian or surrogate multipliers, or to guess the right decisions to make (state-space modification, relaxation of some constraints, etc.).

In line with the challenges highlighted above, we started to address automatic heuristic for dynamic programs with side constraints based on structured-learning techniques. This is a work that had already begun in a collaboration between F. Clautiaux, A. Froger and A. Parmentier (CERMICS). The objective of the project is to determine the best way to apply machine-learning techniques to obtain heuristics via projection, pruning etc. To work on this challenge, we collaborate with Centre Aquitain des Technologies de l'Information et Électroniques (CATIE). Fulin Yan, a student from INSA Rennes, has done an internship with our team on this subject. We are now waiting for a funding answer to propose him a CIFRE PhD thesis.

We additionally started to lay the scientific foundation for a generic framework that embed various techniques from dynamic programming based on regular languages, resource-constrained shortest path, and decision diagrams. We are comparing the different elements coming from the different fields (SAT / MIP / heuristics), determining the common parts and analyzing the main differences between them. Our objective is to develop a common formalism and express the most important methods of the literature using this formalism. This is a work in progress between F. Clautiaux, B. Detienne and A. Froger. We hired Luis Lopes Marques as a PhD student starting from October 2022 to work on this subject. This thesis is financed by our ANR project AD-LIB.

In the near future, we will design exact convergent methods based on aggregation/disaggregation techniques (e.g., use of problem-dependent or primal/dual information) relying on the generic framework cited above. We will focus on finding a systematic way 1) to aggregate an initial model to yield either a relaxation or a restriction and 2) to iteratively disaggregate the current model for obtaining better dual bounds. This is part of our ANR project AD-LIB.

We will also study matheuristics for problems based on time-indexed models. This work will focus on the ability of the method to provide the best possible solution within a given time limit. This is a challenge when solving operational problems where the decision-making time is limited. The key area of research will be (i) how to disaggregate in a way that a good feasible solution can be provided as fast as possible (search strategy) and (ii) how to modify infeasible solutions provided by an aggregated model in order to make them feasible without impairing their cost. This is also a part of the ANR project AD-LIB.

Upon sucessful completion of the above objectives, we may explore many extensions and generalizations to our framework. Among them, we may cite new mechanisms to integrate several different state-space relaxations in the same solution process and extensions to non-linear constraints and objective / dynamic programs based on different semi-rings than $(max,+)$ in ${ℝ}^n$. In the longer term, we wish to explore the links with polyedral theory, and develop extensions of the framework from regular languages to algebraic languages (and to extend our algorithms from graph-based algorithms to hypergraph-based algorithms). Polyhedral theory can be used to improve the performance of our algorithms for solving dynamic programs with side constraints, and to provide the equivalent of the branch-and-cut algorithm for dynamic programs.

3.3 Structure analysis and problem specific studies

Polyhedral analysis 17 remains a part of our project. When integer linear programming methods are involved, information on the structure of the polyhedron representing the feasible region is key for solving hard problems effectively. The most spectacular success of this kind of methods can be found in Concorde 16, a software dedicated to the traveling salesman problem. Network design has also been the subject of many contributions (see e.g. 41). Since we aim to propose methods that can apply to broad classes of problems, we will focus on families of constraints/problems that occur in a large number of practical problems (including capacity, disjunction, precedence or set-covering constraints). There are two promising directions that we can follow in order to study these structures.

In the case where the linear relaxation of an integer programming formulation has an excellent quality, but requests a large computational time, being able to compute good dual solutions rapidly allows to provide useful dual bounds for combinatorial optimization problems. This technique was used for the classical cutting-stock problem under the name dual-feasible functions (see 19). These functions lead to bounds of linear complexity that have an excellent behaviour on average. This concept has already been extended to several packing and location problems (see 53, 63). We plan to explore further extensions to common problem/constraint types.

Many hard problems on graphs become easy when the graph has a special structure 42, typically perfect graphs such as trees, interval or triangulated graphs (see for example 56, 69). In most problems, the graphs do not belong to an easy class. However, one can compute effective relaxations by exhibiting subgraphs of suitable structure in these general graphs. Unfortunately for many classes the problem of finding a subgraph belonging to this class maximizing a certain criterion is NP-hard. We plan to study methods for finding efficient models to compute such relaxations.

In line with the challenges highlighted above, our first objective is to work on polyhedral analysis and computational studies of formulations for the "best" interval subgraph of a general graph. We have identified several families of formulations, among them strong formulations based on an exponential number of variables and/or constraints. Pricing algorithms and separation problems have to be studied in order to efficiently solve these formulations. The same work can be undertaken with chordal graphs, which offer a stronger relaxation (since they are a superclass of interval graphs), for a higher computational cost.

In the longer term, we plan to work on the extension of our work on so-called dual feasible functions to more complex cases. The theory of superadditive and dual-feasible functions have mainly been studied in the case of a single knapsack constraint. Extensions to the intersection of the knapsack polytope with polytopes related to highly structured constraints (such as precedences, or disjunctions) is already a hard challenge.

More generally, we plan to work on several aspects related to polyedral theory: study of extended formulations generated by algebraic grammars. A possible objective is to use implicit formulations expressed in different state spaces to derive efficient cuts for a base formulation ; determining whether considering the structure of the problems will be a stronger tool to derive dual-feasible functions than learning to guess the best dual values from the input parameters (for instance, using structured learning to compute a smaller dual polytope from an initial one) ; studying decomposition schemes based on graph decomposition techniques, such as path or tree-decompositions.

4.1 Integrated problems

Most practical optimization problems are complex, i.e., they involve different types of decisions to be made. Such problems are commonly called integrated. They often involve different time scales related to strategic, tactical and operational decisions. A classical approach to solve such problems is their disintegration into independent problems or stages, where the solution of a stage is the input for next one. We prefer the term "disintegration" here instead of "decomposition" to avoid confusion with decomposition approaches presented above. The disintegration approach usually results in highly sub-optimal solutions. There is large potential for improving the quality of solutions if two and more decision stages are considered together.

Recently there has been an increase in interest for solving integrated optimization problems. Such problems may involve for example integration of production and outbound distribution (production-distribution problem 44), facility location and vehicle routing (location-routing problem 71), inventory management and vehicle routing (inventory routing problem 35), and different levels of distribution (two-echelon routing 57 and cross-docking based distribution).

As we point out above, integrated problems are common in practice but difficult to solve, since they involve synchronization between stages, and different time scales in different stages. Moreover, different classes of problems are usually encountered in different stages. This means that different exact approaches are usually applied to solve such classes of problems. It is often not known how one can efficiently combine these approaches to solve an integrated problem. Thus, heuristic algorithms are used in a vast majority of cases. However, we believe that advances in efficiency and ease of use of exact algorithms and decomposition algorithms in particular allow us to think of applying them here.

Exact approaches are usually limited to small instances of integrated problems, while real-life instances are tackled by heuristic approaches. Nevertheless, for estimating the quality of these heuristics we need approaches that obtain lower bounds of a good quality, even for large scale instances. We think that BCP algorithms are good candidates to obtain such bounds. The column generation approach has however several drawbacks when applied to large-scale integrated problems. When solving pure academic problems, the bottleneck of BCP algorithms is usually the solution of the pricing problem. On the contrary, when dealing with integrated real-life problems, the size of the master problem may become too large and the solution of the restricted master LP becomes a bottleneck. One possible approach to tackle this problem is a dynamic aggregation of constraints in the master LP 31. Another approach could be to use machine learning techniques to choose “good” columns from a large pool generated by the pricing problem. The goal here is to help the column generation algorithm converge faster in the case of a “heavy” master problem or to find “compatible” columns to obtain better primal solutions.

Column generation-based algorithms may also prove useful for obtaining good feasible solutions for integrated problems. A potential directon is to develop matheuristics, i.e., heuristic methods that make use of sophisticated algorithms initially designed for exact methods. The goal is to benefit from the two worlds: exact methods to exploit the special structures of some subproblems, and heuristics to produce solutions in a small amount of time. Column generation-based matheuristics have already shown good results for some integrated problems 18. A common approach is to use heuristics and/or column generation to obtain interesting columns and then solve the restricted master problem as a MIP with a time limit. Then the set of columns can possibly be updated and the restricted master is resolved. However, generic frameworks for such approaches are absent, although they would be highly welcome for a wide class of problems.

Our initial efforts will be concentrated on one or two well-chosen problems of this type. The inventory routing problem (IRP) is probably the most "academic" and simple to state example of integrated problems which stays very hard to solve for both exact and heuristic methods38. In this problem, the decision maker in charge of the delivery is also in charge of the stock level of his/her customer. IRP is widely encountered in practice. The integration of electric vehicles in transportation is also a timely topic. Planning the charging operations of electric vehicles is necessary to account for a wide variety of costs (e.g., time-dependent energy costs), charging infrastructure constraints (e.g., grid restrictions, limited number of chargers), battery degradation, and the potential integration of Vehicle-to-Grid technologies. Taking this variety of constraints into account introduces complex coupling decisions between charging and routing, which leads to integrated problems.

We plan to design a comprehensive modelling and solution framework for planning the charging activities of a fleet of electric vehicles considering time-dependent costs and the potential integration of Vehicle-to-Grid technologies. Particular attention is directed towards decreasing the maximum power of energy required in the planning interval. This will be a joint work with O. Jabali (Politecnico di Milano, Italy) that has already began with a master student from Politecnico di Milano visiting our research team for 4 months last year and studying the case of a fleet of electric buses. Our objectives are to design advanced mathematical methods to produce high-quality solutions in a time efficient manner as well as to propose an exact solution method to solve electric routing problems (E-VRPs) with limited charging infrastructure constraints (based on a previous work 43) using a Branch-and-cut-and-price algorithm. We will investigate the trade-offs that exist between driving time and energy consumption (e.g., existence of alternative paths, speed reduction) either by refining the abstraction of the road network on which E-VRPs are defined or by working directly with the road network.

In the long term, our objective is to be able to cope with an increasing amount of integrated problems, with a focus on supply-chain applications (scheduling, location, inventory, routing, etc.). Our work on this topic will benefit from our findings on new decomposition methods. Once our main objectives are achieved, we will explore methodological tools to take different sources of uncertainty into account (e.g., energy consumption of electric vehicles, time-dependent energy costs) using robust optimization or stochastic programming paradigms.

4.2 Robust optimization

In most decision making problems the data used in the mathematical model is subject to some form of uncertainty. This uncertainty can be caused by measurement errors, variability or the time duration of the processes under study, or simple lack of access to reliable data. Incomplete information can also come from the presence of competitors or adversarial individuals whose policies are not known to the decision maker. Recent events have also shown that the capability to resist to major disasters is an issue for important organizations and critical infrastructure.

There are two commonly accepted paradigms that are used to incorporate uncertainty in mathematical programming: stochastic optimization (including chance-constraints), and robust optimization. In stochastic optimization, one assumes that a probability distribution is known for the uncertain parameters and optimizes a statistical risk measure such as the expected value or the conditional value-at-risk. In this new project, our main tool for dealing with uncertainty will be robust optimization. Unlike stochastic programming, which requires exact knowledge of the probability distributions, robust optimization only describes the variability of the uncertain parameters through bounded uncertainty sets. Further, replacing the risk measures employed in stochastic optimization, with the worst-case realization, robust optimization is more adapted to applications where the decision-making process involves high risks or adversarial participants.

In the last 20 years, the robust optimization field has seen significant progress. Static robust optimization problems, which assume that all decisions are here-and-now, have received considerable attention. They have been formulated and studied with polyhedral, conic and ellipsoidal uncertainty sets. From a complexity viewpoint, it has been shown that these problems are barely harder than their deterministic counterparts in most practical cases. From the numerical viewpoint, mixed-integer linear (or second-order conic) reformulations have been proposed and can handle large-scale problems.

Despite these rather encouraging results, static models suffer from over-conservatism. Indeed, in these models, no recourse action can be taken to change or adapt the solution once the uncertain values have been revealed. This substantially limits the applicability of the robust optimization paradigm since in most applications that involve temporal decision-making, it is possible to adapt the solution to (at least partially) mitigate the effects of uncertainty.

Acknowledging the importance of adjustable models, the scientific community has started to address solution methods for these problems. While it is theoretically well-known that even two-stage linear programs are NP-hard, approximate (affine decision rules) 23, or exact (row-and-column generation algorithms) 22,73 solution methods for problems featuring continuous recourse variables have been proposed in the literature. These two sets of algorithmic tools have made it possible to solve exactly or approximately a large number of adjustable robust optimization problems with continuous recourse. Several studies have also sought to understand the quality of the bounds provided by affine decision rules, as well as proposed extensions to piece-wise affine functions.

The problems we wish to investigate in this project are adjustable robust optimization problems where recourse decisions are integer. The situation is significantly more complex for this setting than for the case of continuous recourse. The two aforementioned approaches do not apply: affine decision rules provide continuous recourse actions by construction, while the separation problem in the row-and-column generation algorithm is based on linear programming duality.

Having no theoretical guarantees regarding their computational complexity, there are some numerical algorithms that aim to solve adjustable robust optimization problems with integer recourse approximately. They are all based on simplifying the type of feasible recourse actions, either by partitioning the uncertainty set or by imposing that the recourse decisions should be chosen among a predetermined finite set of solutions (finite/K-adaptability). Unfortunately, these methods hardly scale up and the recent results 26,48 are limited to small problem instances, which can be partially explained by the fact that they rely on big-$M$ reformulations, known to yield poor continuous relaxations.

In the following years, we plan to develop exact and approximate solution methods for adjustable robust optimization problems with integer recourse. On the one hand, we will study relatively general row-and-column generation algorithms that are based on stronger continuous relaxations than what was done in the literature (aforementioned network formulations and decision diagram-based approaches seem to be promising leads in achieving this goal). On the other hand, we will focus on classes of specific robust models (special cases), which will be studied both from a theoretical and a numerical perspective. Some of our efforts will be spent on improving the efficiency of existing methods such as K-adaptability since these approaches could benefit from the considerable numerical experience of the team.

We developed a first approach to dealing with binary recourse decisions in the context of adjustable robust optimization problems with binary recourse in 20. The main idea of this work is to convexify the recourse feasible region using a Dantzig-Wolfe reformulation. Promising results were presented for the special case where the coupling constraints are deterministic and satisfy additional technical assumptions. In this case, the inner maximization and minimization problems can be interchanged, resulting in a large-scale deterministic equivalent model. We plan to extend this technique to the more general case and to understand how to relax the aforementioned technical assumption.

In the near term, we plan to carry out several projects to respond to the challenges we outline above. We plan to study solution approaches for complex problems arising in future 5G networks. This topic is the subject of a CIFRE thesis supervised by Boris Detienne and Pierre Pesneau in collaboration with Orange. We will also address complex network-design and location problems (topic of ANR project DE-SIDE, in collaboration with Sobolev Institute and Kedge Business School). We will determine easy and hard cases for robust multi-stage network-design problems, depending on the set of constraints, the definition of the uncertainty and the possible second-stage actions, with a focus on integer recourse.

From a methodological perspective, we will adress primal and dual bounds for two- and multi-stage adjustable robust optimization problems (topic of ANR-JCJC project DROI which was granted to Ayse N. Arslan). Building upon the results of 20, we propose to deal with constraint uncertainty in adjustable robust optimization using Lagrange and Fenchel duality, which will provide dual bounds. Several issues need to be resolved in investigating this approach, such as the optimization of the dual (infinite size) problems and how to close the duality gap. We additionally propose to study improved decision rules for these problems.

In the long term, our objectives in relation with this challenge are as follows. First, we want to design efficient approximate approaches to solve multi-stage adjustable robust optimization problems. Although exact methods seem to be out of reach for the current state-of-the-art in the general case, we will also strive to identify special cases which can be solved to exact optimality with numerically viable algorithms. Second, we seek to bridge the gap between multi-stage integer robust optimization and combinatorial games. Some similar concepts are used in both types of problems (typically min-max-min problems have to be solved). This may allow to disseminate our techniques to other fields as well as adapt results obtained for those fields to the context of adjustable robust optimization.

5.1 Impact of research results

Our work 27 on a stochastic generation and transmission expansion planning problem studied at RTE will help the company in their future strategic studies (e.g., studies similar to 65, 64). Specifically, we show how to incorporate the French legislation1 that imposes that the number of hours with energy not served should be in expectation less than or equal to three per year in the mathematical formulation of the expansion planning problem. This work was part of the PhD thesis of Xavier Blanchot.

We are also involved in the PowDev project. The main objective of this project is to evaluate and optimize the resilience of power systems in the context of a massive insertion of renewable energies. The project aims to elaborate a comprehensive and integrated set of decision support tools by considering extreme events in present and future climates, the complexity of the power grid, and socio-economic scenarios.

7.1 New software

8.1 A Continuous Approximation Model for the Electric Vehicle Fleet Sizing Problem

8.2 KidneyExchange.jl: A Julia package for solving the kidney exchange problem with branch-and-price

8.3 A survey on last fifty years of integer linear programming, with a focus on recent practical advances

8.4 Uncertainty reduction in static robust optimization

8.5 A branch-cut-and-price approach for the two-echelon vehicle routing problem with drones

8.6 A Competitive Heuristic Algorithm for Vehicle Routing Problems with Drones

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

8.8 Adjustable robust optimization with objective uncertainty

9.1 Bilateral contracts with industry

We have a collaboration with CATIE (the contract is still being negotiated between Orange and University of Bordeaux), around the PhD thesis of Fulin Yan which started in February 2023. The project focuses on machine learning and optimization.

Participants: François Clautiaux, Aurélien Froger, Fulin Yan.

Together with Inria team INOCS, we are part of the Inria/EDF challenge "Gérer les systèmes électriques de demain". The work package "Algorithmes de décomposition pour les problèmes d'investissement long-terme" focuses on decomposition method for multi-stage integer stochastic programs, with application to capacity planning problems.

Participants: Ayse N. Arslan, Boris Detienne, Aurélien Froger.

Saint-Gobain Research Paris is an industrial research and development centre working for light and sustainable construction of the Saint-Gobain Group, the world leader in light and sustainable construction. The collaboration is centered around the PhD thesis of Pierre Pinet. We are working on stochastic vehicle routing problems with uncertainty.

Participants: François Clautiaux, Ayse N. Arslan, Pinet Pierre.

10.1 International research visitors

10.2 National initiatives

11.1 Promoting scientific activities

11.2 Teaching - Supervision - Juries

11.3 Popularization

12.1 Publications of the year

12.2 Cited publications

• 16 articleD.D .Applegate, W.W .Cook, R.R .Bixby and V.V .Chvatal. Implementing the Dantzig-Fulkerson-Johnson algorithm for large traveling salesman problems.Mathematical Programming, series B972003, 91--153back to text
• 17 articleJ.J .Edmonds. Maximum matching and a polyhedron with 0, 1 vertices.Journal of Research National Bureau of Standards69B1965, 125--130back to text
• 18 articleN.Nabil Absi, D.Diego Cattaruzza, D.Dominique Feillet, M.Maxime Ogier and F.Frédéric Semet. A Heuristic Branch-Cut-and-Price Algorithm for the ROADEF/EURO Challenge on Inventory Routing.Transportation Scienceaccepted2020back to text
• 19 bookC.Clàudio Alves, F.François Clautiaux, J. M.José Manuel Valério De Carvalho and J.Juergen Rietz. Dual-Feasible Functions for Integer Programming and Combinatorial Optimization.SpringerJanuary 2016HALback to text
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