INOCS is a cross-border “France-Belgium” project team in the Applied Mathematics Computation and Simulation Inria domain. The main goal of this team is the study of optimization problems involving complex structures. The scientific objectives of INOCS are related to modeling and methodological concerns. The INOCS team focuses on:

Even if CS problems are in general NP-hard due to their complex nature, exact solution methods or matheuristics (heuristics based on exact optimization methods) are developed by INOCS. The scientific contribution of INOCS will result in a toolbox of models and methods to solve challenging real-life problems.

The research program development of INOCS is to move alternatively:

Even if these two axes are developed sequentially in a first phase, their interactions will lead us to explore them jointly in the mid-term.

An optimization problem consists in finding a best solution from a set of feasible solutions. Such a problem can be typically modeled as a mathematical program in which decision variables must
(i) satisfy a set of constraints that translate the feasibility of the solution and
(ii) optimize some (or several) objective function(s).
Optimization problems are usually classified into strategic, tactical and operational problems, according to the types of decisions to be taken.

We consider that an optimization problem presents a complex structure (CS) when it involves decisions of different types/nature (i.e. strategic, tactical or operational) and/or presents some hierarchical leader-follower structure. The set of constraints may usually be
partitioned into global constraints, linking variables associated with the different types/nature of decision, and constraints involving each type of variables separately. Optimization problems with complex structure lead to extremely challenging problems since a global optimum with respect to the whole sets of decision variables and of constraints must be determined.

Significant progress has been made in optimization to solve academic problems. Nowadays large-scale instances of some Our vision within INOCS is to make the same advances while addressing CS optimization problems. To achieve this goal we aim to develop global solution approaches at the opposite of the current trend. INOCS team members have already proposed some successful methods following this research lines to model and solve CS problems (e.g. ANR project RESPET, Brotcorne et al. 61, 62, Gendron et al. 63, 64, 65, and Strack et al. 66). However, these are preliminary attempts and a number of challenges regarding modeling and methodological issues have still to be met.

A classical optimization problem can be formulated as follows:

In this problem,

The INOCS team plans to address optimization problem where two types of decision are addressed jointly and are interrelated. More precisely, let us assume that variables

In this model,

The INOCS team plans to model optimization CS problems according to three types of optimization paradigms: large scale complex structures optimization, bilevel optimization and robust/stochastic optimization. These paradigms instantiate specific variants of the generic model.

Large scale complex structures optimization problems can be formulated through the simplest variant of the generic model
given above. In this case, it is assumed that

Bilevel programs allow the modeling of situations in which a decision-maker, hereafter the leader, optimizes his objective by taking
explicitly into account the response of another decision maker or set of decision makers (the follower) to their decisions. Bilevel programs are closely related to Stackelberg (leader-follower) games as well as to the principal-agent paradigm in economics. In other words, bilevel programs can be considered as demand-offer equilibrium models where the demand is the result of another mathematical problem.
Bilevel problems can be formulated through the generic CS model when

In robust/stochastic optimization, it is assumed that the data related to a problem are subject to uncertainty. In stochastic optimization, probability distributions governing the data are known, and the objective function involves mathematical expectation(s). In robust optimization, uncertain data take value within specified sets, and the function to optimize is formulated in terms of a min-max objective typically (the solution must be optimal for the worst-case scenario). A standard modeling of uncertainty on data is obtained by defining a set of possible scenarios that can be described explicitly or implicitly. In stochastic optimization, in addition, a probability of occurrence is associated with each scenario and the expected objective value is optimized.

Standard solution methods developed for CS problems solve independent
subproblems associated with each type of variables without explicitly
integrating their interactions or integrating them iteratively in a
heuristic way. However these subproblems are intrinsically linked and
should be addressed jointly. In mathematicaloptimization
a classical approach is to approximate the convex hull of the integer
solutions of the model by its linear relaxation. The main solution
methods are (1) polyhedral solution methods which strengthen this linear
relaxation by adding valid inequalities, (2) decomposition solution
methods (Dantzig Wolfe, Lagrangian Relaxation, Benders decomposition)
which aim to obtain a better
approximation and solve it by generating extreme points/rays. Main
challenges are (1) the analysis of the strength of the cuts and their
separations for polyhedral solution methods, (2) the decomposition
schemes and (3) the extreme points/rays generations for the
decomposition solution methods.

The main difficulty in solving bilevel problems is due to their
nonconvexity and nondifferentiability. Even linear bilevel programs,
where all functions involved are affine, are computationally challenging
despite their apparent simplicity. Up to now, much research has been devoted to
bilevel problems with linear or convex follower problems. In this case, the problem can be reformulated as a
single-level program involving complementarity constraints, exemplifying
the dual nature, continuous and combinatorial, of bilevel programs.

In energy, the team mainly focuses on pricing models for demand side management, on bids definition in the energy market and on the design and pricing of electric car charging stations.

Demand side management methods are traditionally used to control electricity demand which became quite irregular recently and resulted in inefficiency in supply. We have explored the relationship between energy suppliers and customers who are connected to a smart grid. The smart grid technology allows customers to keep track of hourly prices and shift their demand accordingly, and allows the provider to observe the actual demand response to its pricing strategy. We tackle pricing problems in energy according to the bilevel optimization approaches. Some research works in this domain are supported by bilateral grants with EDF.

The increasing number of agents, with different characteristics interacting on the energy market leads to the definition of new types of bidding process. We have modeled this problem as a bilevel one where the lower level is the instance allocating the bids (the ISO).

The proliferation of electric cars in cities has lead to the challenging problem of designing and pricing charging stations in order to smooth the demand over time. We are modeling this problem as a bilevel one where the lower level represents the choice of users in a preference list.

In transportation and logistics, the team addresses mainly integrated problems, which require taking into account simultaneously different types of decision. Examples are location and routing, inventory management and routing or staff scheduling and warehouse operations management. Such problems occur from the supply chain design level to the logistic facility level.

In telecommunications, the team mainly focuses on network design problems and on routing problems. Such problems are optimization problems with complex structure, since the optimization of capacity installation and traffic flow routing have to be addressed simultaneously.

The research works developed in the INOCS team have environmental and societal impacts through the application areas they target. At the environmental level, the works on the optimization of transportation systems aim at reducing the impact of transportation on society. The applied works in energy aim at a better use of the smart grid and the optimization of electricity production from renewable sources. At the societal level, the works developed in the framework of the ANR AGIRE project takes into account musculoskeletal disorders in the activity of employees within a warehouse. Finally, in health, the works conducted on group testing allow the development of effective campaigns of testing of the population in preventive medicine for example.

Group testing is a screening strategy that involves dividing a population into several disjoint groups of subjects. In its simplest implementation, each group is tested with a single test in the first phase, while in the second phase only subjects in positive groups, if any, need to be tested again individually.

To contribute to the effort to tackle the COVID-19 sanitary crisis, we developed this software which allows to create groups of individuals to test via the group testing technique while minimizing a linear combination of the expected number of false negative and false positive classifications.

The test design problem is modeled as a constrained shortest path problem on a specific graph and we design and implement an ad hoc algorithm to solve this problem. We validate the algorithm on instances based on Santé Publique France data on COVID-19 screening tests.

This software is a toolbox that contains algorithms that are frequently used to solve optimization problems tackled by (but not only) the team.

The objective of the toolbox is to contain a set of code skeletons that allow researchers to integrate adequate data structures and basic algorithms for different structures complexity that appears in the optimization problems we study. The current version of the toolbox contains classical heuristic tools (generic local search) to solve, among others, the vehicle rouring problem and its variants. It also contain a code to exactly and heuristically solve the Shortest Path Problem with Ressource Constraints that is usually encountered in the resolution of problem with Branch-and-Price algorithms.

The future objective is to include automatic reformulation tools for bi-level optimization problems and state-of-the-art codes for the development of decomposition methods.

During the year 2023, we addressed different problems/challenges related to the three lines of research: large scale complex structure optimization, bilevel programming and game theory, robust/stochastic programming. The main contributions are summarized in the next sections. In addition, besides these contributions, additional results 12, 15, 17, 18, 26 were obtained which are not discussed hereafter in order to keep the presentation focused on the main achievements.

Energy communities are promising measures aimed at promoting local energy generation and consumption, which are needed to meet the energy transition targets. In 28, the focus is on the optimization of the collective self-consumption rate in energy communities by scheduling members' loads. The community remains connected to the public grid and comprises prosumers, traditional consumers, and distributed storage units. Prosumers can exchange their energy with the public grid or other members. The proposed strategy aims at implementing a Demand Side Management (DSM) program taking advantage of controllable loads' characteristics. A mixed integer linear programming (MILP) formulation of the problem allows, on the one hand, to give the optimal planning for electrical devices' operations. On the other hand, it provides optimal solutions for managing the storage units, peer-to-peer exchanges, and interactions with the public grid to minimize the energy flows from the public grid over time. However, this MILP only allows for solving small problem instances. Thus, we develop a column generation-based heuristic for large problem instances. Our numerical experiments based on real data collected in the south of France show that joining an energy community saves money on energy bills and reduces the total energy drawn from the primary grid by at least

In addition of allowing self-consumption reduction, communties often allows the socialisation of the total cost among its members. In this setting, a recurrent question is related to the design of fair cost allocation mechanisms. In 24, a cooperative is modeled as a business entity whose primary purpose is to provide benefits, services, and goods to its members, who both own and democratically control it. In the context of a cooperative, a fleet typically consists of vehicles owned by self-concerned, individually rational owners who prioritize their own efficiency and the fairness of the system. This fairness refers to how their individual gain aligns with the gain of others. In this paper, we focus on the routing of such cooperative fleets. If we consider only the efficiency of the fleet in terms of minimizing its total cost, the problem studied corresponds to the Multiple Traveling Salesman Problem (MTSP). However, our interest lies in finding both efficient and fair solutions, so we propose two new variants of this problem that integrate and maximize the egalitarian and elitist social welfare. Additionally, to enhance the balance between fleet efficiency and fairness, we propose the systematic elitist and systematic egalitarian social welfare optimization algorithm. Through simulation results, we observe a wide diversity of routes depending on the approach considered. Therefore, a cooperative may choose a model that best balances criteria of efficiency maximization and fairness for its fleet based on its specific requirements.

Vehicle Routing Problems (VRPs) are classical optimization problems in transportation and logistics. In INOCS, we address complex routing problems generalizing the VRP, combining the VRP with another optimization problem, or hybridizing VRP with machine learning approaches. In 23, a MOEA/D framework including a Local Search (LS) based mutation and knowledge discovery mechanisms is the core algorithm used to solve a bi-objective VRP with Time Windows (bVRPTW) where the total traveling cost and the total waiting time of drivers have to be minimized. We enhance the classical LS exploration strategy of the neighborhood from the literature of scheduling and propose new metrics based on customers distances and waiting times to reduce the neighborhood size. We conduct a deep analysis of the parameters to give a fine-tuning of the MOEA/D framework adapted to the LS variants and to the bVRPTW. Experiments show that the proposed neighborhood strategies lead to better performance on both Solomon's and Gehring and Homberger's benchmarks.

Neighborhood search methods are also considered in 47 which tackles Two-level Uncapacitated Facility Location Problems with Single-assignment constraints (TUFLP-S), a problem that arises in industrial applications in freight transportation and telecommunications. First, we recall the real-life application that motivates this research work. Then, we present an integer programming formulation for this variant with modular costs of the TUFLP-S, and we describe an adaptation of variable neighborhood search to the classical TUFLP-S and the variant with modular costs. Focusing on the TUFLP-S, we introduce mathematical programming models based on reformulation techniques and on the relaxation of the integrality of some of the variables associated with location decisions. We then present a Lagrangian relaxation approach for the TUFLP-S, based on solving a single-level uncapacitated facility location problem (UFLP) as the Lagrangian subproblem. We also describe a matheuristic based on a mixed-integer programming-based large neighborhood search. Synthetic computational results on instances derived from the industrial application as well as on large, hard, artificial instances are reported throughout the paper.

An innovative application of the VRP is considered in 29. Inspired by the current practices at JD.com (the largest online retailer by revenue in China), we investigate a delivery problem that we call the traveling salesman problem with bike and robot (TSPBR), where a cargo bike is aided by a self-driving robot to deliver parcels to customers in urban areas. We present two mixed-integer linear programming models and describe a set of valid inequalities to strengthen their linear relaxation. We show that these models can yield optimal solutions of TSPBR instances with up to 60 nodes. To efficiently find heuristic solutions, we also present a genetic algorithm based on a dynamic programming recursion that efficiently explores large neighborhoods. We computationally assess this genetic algorithm on instances provided by JD.com and show that high-quality solutions can be found in a few minutes of computing time. Finally, we provide some managerial insights to assess the impact of deploying the bike-and-robot tandem to deliver parcels in the TSPBR setting.

Order picking is a crucial process in warehouse operations. In a human-operated warehouse, pickers prepare different customer orders. From a managerial perspective, two major decisions need to be made: (i) grouping orders to be collected together by a picker, and (ii) for each picker, determining the route to retrieve the needed items. The Joint Order Batching and Picker Routing Problem (JOBPRP integrates both decisions into a single problem, usually minimizing either total distance or time. In the literature, the JOBPRP usually assumes a situation without any congestion caused by the pickers, however unrealistic when there are many pickers in a warehouse. In practice, congestion causes inefficiency, increases costs, reduces performance, and leads to accidents. To estimate congestion levels, we divide the planning horizon of the picking activity into different time intervals and timing variables are introduced. If two or more pickers are in the same sub-aisle during the same time interval, a delay in travel time is imposed. Because picking activities are performed by humans, practical considerations must be taken into account when defining a feasible picking route, e.g. a picker cannot unnecessarily wait and cannot follow complicated or long paths between two consecutive picking positions. To solve the JOBPRP including the effect of picker congestion, an extended mathematical formulation is presented, with the main objective of minimizing the total time, including delay. A heuristic solving approach is proposed based on solving of the linear relaxation of the formulation by an exact column generation procedure. In each iteration, a negative reduced column is produced using a dedicated labelling algorithm, exploring routes associated to congestion level for each sub-aisle and time interval. To evaluate the modelling and the performance of the solutions provided by the proposed approach, a discrete event simulation tool is developed. Several experiments are performed to compare the results with optimal JOBPRP solutions not considering congestion 45.

The Picker Routing Problem (PRP), which consists in finding a minimum-length tour between a set of storage locations in a warehouse, is one of the most important problems in the warehousing logistics literature. Despite its popularity, the tractability of the PRP in conventional multi-block warehouses remains an open question. The technical note in 58 aims to fill this research gap by establishing that the PRP is strongly NP-hard.

We propose a survey 20 on vehicle routing problems with multiple commodities. In most routing problems, only one commodity is explicitly considered. This may be due to the fact that, indeed, a single commodity is involved, or multiple commodities are transported, but they are aggregated and modeled as a single commodity, as no specific requirement imposes their explicit consideration. However, there exist cases in which this aggregation is not possible due to the characteristics of the commodities or to the fact that it would lead to sub-optimal routing plans. The survey focuses on the analysis of the settings of the problems and the features of the commodities that require explicit consideration of disaggregated commodities in routing problems. We show that problem settings are inherently different with respect to the single commodity problems, and this has a consequence on both models and solution approaches, which cannot be straightforwardly adapted from the single commodity cases. We propose a classification of the routing problems with multiple commodities and discuss the motivations that force considering the presence of multiple commodities explicitly. Specifically, we focus on the modeling perspective by proposing a general formulation for routing problems with multiple commodities and showing how this formulation can be adapted to the different features that characterize the problem classes discussed in the survey. Also, for each major class of problems, promising future research directions are discussed by analyzing what has been studied in the current literature and focusing on challenging topics not covered yet.

We address two specific routing problems with multiple commodities. The commodity constrained Split Delivery Vehicle Routing Problem (C-SDVRP) is a routing problem where customer demands are composed of multiple commodities. A fleet of capacitated vehicles must serve customer demands in a way that minimizes the total routing costs. Vehicles can transport any set of commodities and customers are allowed to be visited multiple times. However, the demand for a single commodity must be delivered by one vehicle only. In this work, we developed a heuristic with a performance guarantee to solve the C-SDVRP. The proposed heuristic is based on a set covering formulation, where the exponentially-many variables correspond to routes. First, a subset of the variables is obtained by solving the linear relaxation of the formulation by means of a column generation approach which embeds a new pricing heuristic aimed to reduce the computational time. We test the heuristic algorithm on benchmark instances from the literature. The comparison with the state-of-the-art heuristics for solving the C-SDVRP shows that our approach significantly improves the solution time, while keeping a comparable solution quality and improving some best-known solutions. In addition, our approach is able to solve large instances with 100 customers and six commodities, and also provides very good quality lower bounds 57.

A extended variant of the C-SDVRP is the Multi-Commodity two-echelon Distribution Problem (MC2DP). In the MC2DP, multiple commodities are distributed in a two-echelon distribution system involving suppliers, distribution centres and customers. Each supplier may provide different commodities and each customer may request several commodities as well. In the first echelon, capacitated vehicles perform direct trips to transport the commodities from the suppliers to the distribution centres for consolidation purposes. In the second echelon, each distribution centre owns a fleet of capacitated vehicles to deliver the commodities to the customers through multi-stop routes. Commodities are compatible, i.e., they can be mixed in the vehicles. Finally, customer requests can be split by commodities, that is, a customer can be visited by several vehicles, but the total amount of each commodity has to be delivered by a single vehicle. The aim of the MC2DP is to minimise the total transportation cost to satisfy customer demands. We propose a set covering formulation for the MC2DP where the exponential number of variables relates to the routes in the delivery echelon. We develop a Branch-Price-and-Cut algorithm (BPC) to solve the problem. The pricing problem results in solving an Elementary Shortest Path Problem with Resource Constraints (ESPPRC) per distribution center. We tackle the ESPPRC with a label setting dynamic programming algorithm which incorporates ng-path relaxation and a bidirectional labelling search. Pricing heuristics are invoked to speed up the procedure. In addition, the formulation is strengthened by integrating capacity cuts and two families of valid inequalities specific for the multiple commodities aspect of the problem. Our approach solves to optimality 439 over the 736 benchmark instances from the literature 60.
The optimality gap of the unsolved instances is

Warehouses are the scene of complex logistic problems integrating different decision layers. This work addresses the Joint Order Batching, Picker Routing and Sequencing Problem with Deadlines (JOBPRSP-D) in rectangular warehouses. To tackle the problem an exponential linear programming formulation is proposed. It is solved with a column generation heuristic able to provide valid lower and upper bounds on the optimal value. We start by showing that the JOBPRSP-D is related to the bin packing problem rather than the scheduling problem. We take advantage of this aspect to derive a number of valid inequalities that enhance the resolution of the master problem. The proposed algorithm is evaluated on publicly available data-sets. It is able to optimally solve instances with up to 18 orders in few minutes. It is also able to prove optimality or to provide high-quality lower bounds on larger instances with 100 orders 39, 49. To the best of our knowledge this is the first work that provides optimality guarantee on large size instances for the JOBPRSP-D, the results can therefore be used to assert the quality of heuristics proposed for the same problem.

Feature selection is a fundamental process to avoid overfitting and to reduce the size of databases without significant loss of information that applies to hierarchical clustering. Dendrograms are graphical representations of hierarchical clustering algorithms that for single linkage clustering can be interpreted as minimum spanning trees in the complete network defined by the database. In 21, we introduce the problem that determines jointly a set of features and a dendrogram, according to the single linkage method. We propose different formulations that include the minimum spanning tree problem constraints as well as the feature selection constraints.

In 11, we study a discrete version of the classical classification problem in Euclidean space, to be called the geodesic classification problem. It is defined on a graph, where some vertices are initially assigned a class and the remaining ones must be classified. This vertex partition into classes is grounded on the concept of geodesic convexity on graphs, as a replacement for Euclidean convexity in the multidimensional space. We propose two new integer programming models along with branch-and-bound algorithms.

The massive integration of renewable energy and the growing involvement of prosumers have changed the energy landscape, calling for an urgent restructuring of the electricity market. Game theory is a powerful tool to analyse the outcome of situation of conflicts involving a large number of heterogeneous agents with selfish interest. In this setting, we propose a new forward electricity market framework that admits heterogeneous market participants with second-order cone strategy sets, who accurately express the nonlinearities in their costs and constraints through conic bids, and a network operator facing conic operational constraints. In contrast to the prevalent linear-programming-based electricity markets, we highlight how the inclusion of second-order cone constraints improves uncertainty-, asset-, and network-awareness of the market, which is key to the successful transition towards an electricity system based on weather-dependent renewable energy sources. We analyze our general market-clearing proposal in 27 using conic duality theory to derive efficient spatially-differentiated prices for the multiple commodities, comprised of energy and flexibility services. Under the assumption of perfect competition, we prove the equivalence of the centrally-solved market-clearing optimization problem to a competitive spatial price equilibrium involving a set of rational and self-interested participants and a price setter. Finally, under common assumptions, we prove that moving towards conic markets does not incur the loss of desirable economic properties of markets, namely market efficiency, cost recovery, and revenue adequacy. Our numerical studies focus on the specific use case of uncertainty-aware market design and demonstrate that the proposed conic market brings advantages over existing alternatives within the linear programming market framework.

Information sharing and coordination mechanisms impact the properties of the resulting market equiliria. Five market models for the procurement of flexibility by Transmission (TSO) and Distribution System Operators (DSOs), based on several TSO-DSO coordination schemes, including a disjoint distribution, disjoint transmission, common, fragmented, and multi-level market are proposed in 25. The properties of these models are then analyzed. In particular, the common market is first proven to be more efficient than the other market models. Then, different methods are proposed to adequately price TSO/DSO interface flows, when procuring cross-grid flexibility. It is then shown that, when interface flows are optimally priced, the fragmented and multi-level market solutions converge to those of the common market, reaching optimal efficiency. To prevent the need for any network information sharing among system operators in the different coordination schemes, decomposition methods based on bi-level programming and the Alternating Direction Method of Multipliers (ADMM) are proposed. A developed case study, considering an interconnected transmission-distribution system, corroborates the mathematical findings by highlighting the greater efficiency of the common market, the effect of adequate interface pricing on reducing procurement costs, and the capability of the decomposition methods to reach optimal market solutions with limited information exchange.

The duality results from standard optimization setup that we applied to noncooperative games in 25 can be extended to strategic pricing and resource allocation problems for communication networks. In 16, we consider a marketplace in the context of 5G network slicing, where Application Service Providers (ASP), i.e., slice tenants, providing heterogeneous services, are in competition for the access to the virtualized network resource owned by a Network Slice Provider (NSP), who relies on network slicing. We model the interactions between the end users (followers) and the ASPs (leaders) as a Stackelberg game. We prove that the competition between the ASPs results in a multi-resource Tullock rent-seeking game. To determine resource pricing and allocation, we devise two innovative market mechanisms. First, we assume that the ASPs are pre-assigned with fixed shares (budgets) of infrastructure, and rely on a trading post mechanism to allocate the resource. Under this mechanism, the ASPs can redistribute their budgets in bids and customise their allocations to maximize their profits. In case a single resource is considered, we prove that the ASPs' coupled decision problems give rise to a unique Nash equilibrium. Second, when ASPs have no bound on their budget, we formulate the problem as a pricing game with coupling constraints capturing the shared resource finite capacities, and derive the market prices as the duals of the coupling constraints. In addition, we prove that the pricing game admits a unique variational equilibrium. We implement two online learning algorithms to compute solutions of the market mechanisms. A third fully distributed algorithm based on a proximal method is proposed to compute the Variational equilibrium solution of the pricing game. Finally, we run numerical simulations to analyse the market mechanism's economic properties and the convergence rates of the algorithms.

We formulate a security game in a context of mixed armament acquisition, involving a finite set of Nations in strategic relationship, with utility functions which are nonlinear and non-differentiable on the boundary of their sets of definition. Since we want to study the long-term effect of the Nations' investment in nuclear weapons, we focus on the steady-state analysis of the game. This requires us to extend the classical results from Rosen on compact-convex games, to unbounded convex games, relying on the coercivity property of the Nations' utility functions. In addition, we prove the existence and uniqueness, under mild assumptions, of an interior point Nash Equilibrium solution of the game. Simulations are performed in case of a duopoly, highlighting the efficiency loss reduction and stabilizing effect of nuclear armaments by comparison with the conventional-only setting 32, 55.

Item and set orderings help with data management. Depending on the context, it is just as important to order a list of items (customers from different provinces, companies from different sectors, players from different teams) as it is to order a list of sets of these items (provinces, sectors, teams). It is evident that the order that is chosen for the items is not independent of the order that is chosen for the sets. It is possible that several set orders are sensible for the same item order and vice versa, that several item orders are sensible for the same set order. In 22, we propose a bilevel model to calculate an adequate order of items when an order of sets is available and another bilevel model to calculate an adequate order of sets when an order of items is available. In addition, it is shown how to reduce both bilevel models to single level models.

In 19, we consider resilience of service networks that are composed of service and control nodes to node‐targeted attacks. Two complementary problems of selecting attacked nodes and placing control nodes reflect the interaction between the network operator and the network attacker. This interaction can be analyzed within the framework of game theory. Considering the limited performance of the previously introduced iterative solution algorithms based on non‐compact problem models, new compact integer programming formulations of the node attack optimization problem are proposed, which are based on the notion of pseudo‐components and on a bilevel model. The efficiency of the new formulations is illustrated by the numerical study that uses two reference networks (medium‐size and large‐size), and a wide range of the sizes of attacks and controllers placements.

Cloud computing has transformed numerous sectors of businesses, governments, and individuals’ lives with its benefits of processing power, resource simplicity, scalability, and accessibility. Predictions anticipate a significant upsurge in global data traffic by 2025, driven by an increase in demand for cloud computing. Although cloud computing is energy-efficient compared to private computers, it is still wasteful. Data centers often underutilize servers, which continue to consume considerable amounts of energy even when idle. The inefficiency originates in the overstocking of computers due to consumer behavior and service-level buffering.1 To face this problem, we propose a cloud sharing system to enhance resource efficiency. The system’s framework is formulated as a mixed integer bilevel problem, which features two distinct follower types representing the different types of customers in the public cloud, namely, long-term consumers with monthly subscriptions and short-term consumers who seek on-demand access. The cloud service manager’s (CSM) goal, operating as the leader, is to incentivize long-term consumers to share their resources through rewards to allocate shortterm consumers without requiring new resources. Resources energy consumptions measures are included in the objective functions of the leader. We have defined new cuts and have developped a branch and cut algorithm to solve the problem.

We consider a provider of electric vehicle charging that operates a network of charging stations and wishes to use dynamic pricing to maximize profit and reduce the impact on the electric grid. We propose a bilevel model with a single leader and multiple disjoint followers. The customers are followers, and each makes decisions independently from the others. The provider is the leader, sets the prices for each station at each time slot, and ensures there is enough energy to charge. The behavior of each customer is represented by the combination of a preference list of (station, time) pairs and a reserve price. In this way, the proposed model accounts for the heterogeneity of customers with respect to price sensitivity and charging preferences. We solve the bilevel optimization problem using a reformulation based on optimality conditions and compare our results to other approaches in the literature. The proposed reformulation is able to solve large-scale instances, and the computational results show that the maximum consumption peak can be significantly reduced with only a slight degradation of the provider's profit.

In the same spirit another work is related to the charging of autonomous electrical cars. A special attention is devited to the interactions between the transportation and electrical grid networks. A new model based on an extension of the pick-up and delivery problem is proposed.

The overall objective of the EuropanSUM project is to facilitate the transformation of mobility in 15 European cities by 2026 and in 30 European cities by 2030. This transformation involves the integration of new shared mobility modes with public transport, focusing on innovation, interconnectivity, environmental sustainability, safety, resilience and replicability. The aim is to ensure the affordability and reliability of these modes for end users, while maintaining financial sustainability that strengthens the competitiveness of European businesses.

In this setting we develop new bilevel joint pricing models to incetivize the users to combine new shared mobility modes with public transportation. Several path offers will be provided to the users.

INOCS has recently started a research track on the study of Multi-Agent Reinforcement Learning (MARL) in stochastic networks of agents. Within this area, our goal is to study the dynamics of electricity markets involving multiple competitive generators through MARL approaches. We start by formulating the electricity market as a two-stage stochastic game, involving a finite set of conventional and renewable energy producers, which bid on the day-ahead market, and an Independent System Operator (ISO), which is responsible for the clearing of the market. We assume that a predetermined part of the producers are non-strategic, bidding at their marginal costs, while the others might bid strategically trying to learn the outcome of the clearing. In the first stage, the strategic producers optimize simultaneously their bids by minimizing their expected costs (opposite of the expected profits), which is the difference between their production cost and the payment they receive from the ISO. The renewable energy producers' objective functions include a penalty assigning a cost to the imbalances caused by their forecast errors. In the second stage, the ISO receives the bids of all the producers. It clears the market by determining the activated volumes for each producer and a price minimizing the total cost under capacity constraints, including a conditional value at risk (CVaR) constraint for the renewable producers, capturing the risk aversion level that the requested volume violates their uncertain capacity. We derive closed form expressions for the producers' best-responses considering pay-as-clear and pay-as-bid as pricing schemes, and simulate the market dynamics, using MARL. To that purpose, we rely on modified versions of two actor-critic algorithms: Deep Deterministic Policy Gradient (DDPG) and Soft Actor-Critic (SAC). The simulations show how the producers adapt dynamically their strategies to learn the best bidding strategy, under limited information exchange. Finally, we identify conditions for the convergence of MARL algorithms to local equilibria of the stochastic game. First results in this area have been presented in 35, 41.

MARL is a widely researched technique for decentralized control in complex large-scale systems with interacting agents. When predictions support decisions they may influence the outcome they aim to predict. We call such prediction performative, i.e., the prediction influence the target. A conceptual novelty is an equilibrium notion called performative stability in the literature. We apply this new concept to a peer-to-peer electricity market modeled as a network game, where End Users (EUs) minimize their cost by computing their demand and generation while satisfying a set of local and coupling constraints. Their nominal demand constitutes sensitive information, that they might want to keep private. We prove that the network game admits a unique Variational Equilibrium (VE), which depends on the private information of all the EUs. A data aggregator is introduced, which aims to learn the EUs’ private information. The EUs might have incentives to report biased and noisy readings to preserve their privacy, which creates shifts in their strategies. Relying on performative prediction, we define a decision-dependent game

In 37, we examine the impact of the coupling between an introduced data market, in which agents can purchase a forecast of their renewable energy sources' generation levels to improve their estimation quality, and a peer-to-peer electricity market, enabling prosumers to trade energy in a decentralized manner with their peers, amidst the growing trend of decentralization and uncertainty of Renewable Energy Sources (RES) in electricity markets. The study formulates the P2P trading as a generalized Nash equilibrium problem and identifies conditions for achieving a maximized efficiency of the peer-to-peer electricity market, one of which is prosumers' participation in the forecast market. Along these lines, the analysis demonstrates that prosumers have incentives to participate in the forecast market and outlines the conditions, considering the case in which the forecasts are given in the form of a Gaussian distribution. Numerical examples using Pecan Street data demonstrate the theoretical findings and provide illustrations for the general case, as well as highlight the mutual benefits of market coupling for forecast sellers and electricity market agents.

We consider mixed-integer linear quantile minimization problems that yield large-scale problems that are very hard to solve for real-world instances. We motivate the study of this problem class by two important realworld problems: a maintenance planning problem for electricity networks and a quantile-based variant of the classic portfolio optimization problem. For these problems, we develop valid inequalities and present an overlapping alternating direction method. Moreover, we discuss an adaptive scenario clustering method for which we prove that it terminates after a finite number of iterations with a global optimal solution. We study the computational impact of all presented techniques and finally show that their combination leads to an overall method that can solve the maintenance planning problem on large-scale real-world instances provided by the EURO/ROADEF challenge 2020 and that they also lead to significant improvements when solving a quantile-version of the classic portfolio optimization problem 14.

In 31, we present a novel algorithmic approach to optimize traffic engineering in segment routing networks, accounting for demand uncertainty. In particular, we propose a stochastic approach to online segment routing which uses a conditional value at risk when accounting for the traffic matrix uncertainty. This approach can perform significantly better than the worst-case approach often considered in the literature. We also show that depending on the demand volatility, our stochastic approach can be further optimized in that it is sufficient to account for only a part of the demand without sacrificing traffic engineering quality.

Program PGMO funded by the Fondation Mathématiques Jacques Hadamard and EDF R&D. Stackelberg Games for Flexibility (Dis)Aggregation (2022 – 2024).

BIO-SEL

Title: Bilevel Optimization in Security, Energy and Logistics

Duration: 2020–2023

PIs: Martine Labbé (Université Libre de Bruxelles), Vladimir Marianov (PUC Chile)

Partners: Pontificia Universidad Catolica de Chile Santiago (Chile)

Summary: This projet is devoted to bilevel optimization problems with application in the security,
energy, and logistics domains. Stackelberg games, including one defender and several followers,
bidding problems in energy supply markets and product selection problems will be considered.
Mixed integer optimization models and efficient algorithms to solve them will be developed.

GALANGAL

Title: Game Theoretic Learning and Optimization for Networked Electricity Markets

Duration: 2023–2025

PIs: Hélène Le Cadre (Inria), Michel Gendreau (Polytechnique Montréal, Canada)

Partners: Polytechnique Montréal (Canada), Edinburgh University (Scotland)

Summary: Game Theory is the study of interacting decision makers. A large part of the work in this area has focused on equilibrium computation, but another relevant question is how agents might reach an equilibrium, especially given that no single agent has full information on the state of the system or full authority over the strategies of the other agents. GALANGAL research directions are split into three axes: (i) learning market equilibria preserving statistical privacy using performative prediction, (ii) deep reinforcement learning to control interconnected buildings, (iii) equilibrium tracking, towards robust markets.

SOGGA

Title: Stochastic Optimization, Generalized Games and Applications

Duration: 2024–2025

Coordinator: Dider Aussel (Université de Perpignan, France)

Partners: University of Perpignan, Universidad de Chile and Universidad de O'Higgins (Chile), Universidad del Pacifico (Peru), Inria Lille

Summary: SOGGA proposes to focus in four theoretical research lines: (i) continuity-like properties in equilibrium problems, (ii) regularity in generalized equilibrium problems, (iii) bilevel games with decision-dependent uncertainty (Luce Brotcorne), (iv) algorithms and mechanism design in learning games (Hélène Le Cadre).

CityFreight – Freight Logistics in Sustainable Cities

Pamela Bustamante

Sophia Calderon Pimienta

Clara Chini Nielsen

Alejandro Jofre

Walter Rei

SUM

Title: Seamless Shared Urban Mobility

Duration: 2023–2026

Partners: Inria INOCS (project coordinator), Delft University (The Netherlands), NTUA, Ertico, Polis, Vedecom, Technical University of Munich, City of Munich, Municipality of Pentali, sixt, Freenow, Krakow Municipality, Jagiellonian University, Ret, Fredrikstad Municipality, Nextbike Cyprus, Chalmers University of Technology, MOBYx, Cunisa, Coimbra Municipality, ZF, TU Coimbra, Jerusalem Municipality, Tel Aviv University, Sigma6, Larnaca Public Transport, Siemens Mobility B.V., HYKE, University of Twente, TPG.

Summary: The overall objective of the SUM project is to facilitate the transformation of mobility in 15 European cities by 2026 and in 30 European cities by 2030. This transformation involves the integration of new shared mobility modes with public transport, focusing on innovation, interconnectivity, environmental sustainability, safety, resilience and replicability. The aim is to ensure the affordability and reliability of these modes for end users, while maintaining financial sustainability that strengthens the competitiveness of European businesses.

CHIST–ERA SEC–OREA

Title: Supporting Energy Communities – Operational Research and Energy Analytics

Duration: 2022–2025

Partners: Inria INOCS, Université Libre de Bruxelles (Belgium), University College of Dublin (Ireland), Riga Technical University (Lithuania)

Summary: SEC–OREA aims to enable local energy communities (LECs) to participate in the decarbonisation of the energy sector by developing advanced efficient algorithms and analytics technologies.
The SEC–OREA projects has enabled the hiring of two Ph.D. students within INOCS: Juan Sepulvea and Cristian Aguayo.

ANR project AGIRE (2020-2024): "Decision system for smart management of resources in warehouses" in collaboration with Ecole des Mines de Saint-Etienne (Gardanne), Université Gustave Eiffel (Champs-sur-Marne), HappyChic (Tourcoing).
This project addresses human resources management in warehouses which supply either sale points (B2B) or final consumers (B2C). Nowadays, such warehouses are under pressure. This is mainly due to the no inventory policy at the sale points and to the constant growth of e-commerce sales in France and Europe. In terms of logistics, this translates into an increasing number of parcels to prepare and to ship to satisfy an order, which is known typically a few hours before. Moreover, the total number of products to be packed varies very significantly from day-to-day by a factor of at least 3.
The novelty of the project is twofold:
(1) The human factor is explicitly be taken into account. It is integrated in the mathematical models and algorithms that are developed for the project. The aim is to improve the quality of employees' work ensuring the efficiency of the logistic system;
(2) Problems at different decision levels are integrated and tackled jointly. At the tactical level, the main issues are workload smoothing and the management of the storage zone. At operational level, the major issues concern the rearrangement of the picking zone, the picking tours, and the dynamic reorganization of activities to manage uncertainties.

ANR project ADELE (2022-2025): “Resource Allocation in City Logistics with Demand Uncertainty” in collaboration with LCOMS (Univ. of Lorraine), Toulouse Business School, Colisweb. A central issue in city logistics (CL) is to design logistics systems that move goods to, from, and within urban areas while meeting sustainability goals. A central role is played by the orchestrator. The orchestrator is the stakeholder that operates and organizes a CL system when multiple stakeholders are implied.
In ADELE, we tackle the planning problem faced by the orchestrator in coordinating and managing the resources offered by carriers or logistics service providers. The problem aims to determine what logistics facilities should be used and when and where the vehicles of the carriers should be assigned to cover the demand in the most efficient way. A key feature is that demand is uncertain. We consider two main variants depending on whether the CL system is one or two tiers. ADELE aims to develop new efficient mathematical models and decision support methods. we aim to design and implement ad-hoc optimization algorithms based on mathematical modeling. This project is a continuation of the INRIA Innovation Lab Colinocs.

STaRS project SITAR (2022-2025): The SITAR project is funded by the Région Hauts-de-France. The research within SITAR is split into two axes:
(1) modeling and analysis of an optimal taxation problem, involving sanctions, formulated as a Stackelberg game involving a partial equilibrium problem at the lower level. The set of equilibria will be characterized analytically. Different settings with multi-commodity competition and different out-neighbors communication patterns will be considered. Applications will be made to the gas market.
(2) development and performance analysis of learning algorithms for equilibrium tracking. A fundamental open problem is game theory is the computation of a specific equilibrium among all the possible ones, e.g., the optimal one with respect to (the possible combination of) different criteria, e.g., efficiency loss minimization, accuracy of the equilibrium approximation maximization, convergence rate minimization. The existing algorithms have convergence guarantees toward an arbitrary, possibly inefficient (with respect to those criteria), equilibrium. We aim to contribute to this second axis by deriving distributed algorithms for the computation of an optimal equilibrium in noncooperative games, and extend the results to the time-varying setting to track the sequence of optimal equilibria.
The SITAR project has enabled the hiring of a 18 months postdoc researcher, Fedy Pokou, who will join the INOCS Team in February 2024.

Annals of Operations Research, Applied Computing and Informatics, Central European Journal of Operations Research, Computers & Operations Research, Computational Optimization and Applications, Discrete Applied Mathematics, EURO Journal on Transportation and Logistics, European Journal of Operational Research, IISE Transactions, INFORMS Journal on Computing, International Journal of Management Science and Engineering Management, International Transactions in Operational Research, Journal of Optimization Theory and Applications, Mathematical Programming Computation, Networks, Omega, Operations Research, Optimization and Engineering, RAIRO - Operations Research, Transportation Science, IEEE Transactions on Automatic Control: Luce Brotcorne, Diego Cattaruzza, Bernard Fortz,Martine Labbé, Hélène Le Cadre, Maxime Ogier, Frédéric Semet.

Multi-Agent Reinforcement Learning for Strategic Bidding in Two Stage Electricity Markets. IMACS23 – 21st IMACS World congress, Rome, Italy, Sep 11, 2023: Francesco Morri.