INOCS is a new 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. In short this project team is called INOCS for INtegrated Optimization with Complex Structures. The scientific objectives of INOCS are related to modeling and methodological concerns. The INOCS team will focus on:

integrated models for problems with complex structure (CS) taking into account the whole structure of the problem;

on the development of solution methods taking explicitly into account
*the nature and the structure of the decisions as well as the
properties of the problem*.

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) will be 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 :

*from problems towards new approaches in optimization*: Models
and solution algorithms will be developed to fit the structure and
properties of the problem. From them, new generic approaches will be
used to optimize problems with similar properties.

*from innovative approaches towards problems*: The relevance of
the proposed approaches will be assessed by designing new models
and/or solution methods for various classes of problems. These models
and methods will be based on the extension and integration of
specific, well studied, models and methods.

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

satisfy a set of constraints that translate the feasibility of the solution and

optimize some (or several) objective function(s). Optimization problems are usually classified according to types of decision to be taken into strategic, tactical and operational problems.

We consider that an optimization problem presents a complex structure when it involves decisions of different types/nature (i.e. strategic, tactical or operational), and/or presenting 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 a 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 progresses have been made in optimization to solve academic
problems. Nowadays large-scale instances of some NP-Hard problems are
routinely solved to optimality. *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.* 2011, 2012,
Gendron *et al.* 2009, Strack *et al.* 2009). 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,

INOCS team plan 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 his/her 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
sub-problems 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 *mathematical* *optimization*
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 i) polyhedral solution methods which strengthen this linear
relaxation by adding valid inequalities, ii) 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 i) the analysis of the strength of the cuts and their
separations for polyhedral solution methods, ii) the decomposition
schemes and iii) the extreme points/rays generations for the
decomposition solution methods.

The main difficulty in solving *bilevel problems* is due to their
non convexity and non differentiability. 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.

It is hard to find an aspect of our modern-day economy whose design, management and control do not critically depend on the solution of one or more CS decision problems. Even if they are pervasive, many of them are still not “satisfactorily” solved and constitute a strong challenge to research teams nowadays. The innovative research goals of INOCS have, without doubt, a strategic importance in the application field. CS problems appear in a broad range of application fields such as the next one cited hereafter.

*the energy sector* where decisions of distinct nature such as
production and distribution are jointly determined;

*supply chain management* where location and routing decisions
have to be defined jointly even if they refer to different time
horizons;

*revenue management* where the determination of prices for
services or products requires to take explicitly into account the
strategic consumers' behaviour.

Industrial contract with EDF, Bilevel models for tariff setting problems in the energy field (2010-2011; 2012-2015)

Industrial contract with Coliweb, Load charge assignent for freight deliveries (2015-2016)

Gaspard Monge Program for Optimisation and operationnal research, Design and Pricing of Electricity Services in a Competitive Environment (2015-2018)

Gaspard Monge Program for Optimisation and Operationnal Research, BENMIP A Generic Benders Decomposition based (Mixed) Integer Programming Solver, (2015-2016)

ANR project Transports Terrestres Durable “RESPET - Gestion de réseaux de service porte-à-porte efficace pour le transport de marchandises”, in collaboration with LAAS (Toulouse), DHL, JASSP, LIA (Univ. Avignon) (2011-2015).

Combinatorial Optimization: Meta-heuristics and Exact Methods (2012-2017, coordinator: Bernard Fortz (GOM-ULB/INOCS-Inria). Study and modeling of combinatorial optimization problems; Advancements in algorithmic techniques; Implementation of solution methods for large-scale, practically relevant problems.

Program: BEWARE FELLOWSHIPS Academia

Project acronym: PARROT

Project title: Planning Adapter performing ReRouting and Optimization of Timing

Duration: 10/2014 - 09/2017

Coordinator: Martine Labbé (ULB)

Other partners: INFRABEL (Belgique).

Abstract: The Belgian railway company needs a new tool for the trains which have to be rescheduled when the company must do some maintenance operations on the network. The difficulties are the number of constraints, the size of the network, the quantity of trains and many other features related to the Belgian railway system. These difficulties imply that some choices have to be made to balance the quantity of work feasible in the 3 years project. After developing an interface between the INFRABEL database and the framework used in this project, a first model (MIP) will be implemented and then tested.

Program: JPI Urban Europe

Project acronym: e4-share

Project title: Models for Ecological, Economical, Efficient, Electric Car-Sharing

Duration: 11/2014 - 10/2017

Coordinator: Markus Leitner (U. Vienna, Austria)

Other partners:

AIT, Vienna, Austria

GOM, Université Libre de Bruxelles (Inria/INOCS)

Department of Electrical, Electronics and information Engineering, Alma Mater University of Bologna, Italy

iC consulenten Ziviltechniker GesmbH, Vienna, Austria

Abstract: Car-sharing systems and the usage of electric cars become increasingly popular among urban citizens. Thus, providing vast opportunities to meet today�s challenges in terms of environmental objectives, sustainability and living quality. Our society needs to manage a transformation process that ultimately shall lead to fewer emissions and less energy consumption while increasing the quality of public space available.

In e4-share, the team will lay the foundations for efficient and economically viable electric car-sharing systems by studying and solving the optimization problems arising in their design and operations. A main goal is to derive generic methods and strategies for optimized planning and operating in particular for flexible variants which best meet preferences of customers but impose nontrivial challenges to operators. This project will develop novel, exact and heuristic, numerical methods for finding suitable solutions to the optimization problems arising at the various planning levels as well as new, innovative approaches considering these levels simultaneously.

The project e4-share (Models for Ecological, Economical, Efficient, Electric Car-Sharing) runs from October 2014 to October 2017 and is funded by FFG, INNOVIRIS and MIUR via Joint Programme Initiative Urban Europe. The project comprises an interdisciplinary team of five partners from Austria, Belgium and Italy.

CIRRELT, GERAD, Montreal (P. Marcotte, G. Savard, M. Gendreau, G. Laporte, B. Gendron, ..)

University of Maastricht (Stan Van Hoesel)

Politecnico di Milano (Edouardo Amaldi)

University of Lisbon (Luis Gouveia)

University of Aveiro (Cristina Requejo)

University of Sevilla (Justo Puerto)

University of Chile (Fernando Ordonez)

PhD in progress : Luciano Porretta, “Models and methods for the study of genetic associations”, May 2011, Bernard Fortz

PhD in progress : Sezin Afsar,"Revenue Optimization and Demand Response Models using bilevel programming in smart grid systems”,October 2011 , Luce Brotcorne

PhD in progress : Martim Moniz, “Traffic engineering in Ethernet networks”, November 2012, Bernard Fortz and Luis Gouveia

PhD in progress : Bayrem Tounsi,"Gestion de rśeaux de services porte à porte pour le transport de marchandies ”, October 2012 , Luce Brotcorne

PhD in progress : Carlos Casorran, “A Mathematical Optimization Approach for Stack- elberg Solutions of Bimatrix Games", July 2013, Martine Labbé

PhD in progress : Fabio Sciamannini, “Exact algorithms for variants of the coloring problem”, September 2014, Bernard Fortz, Martine Labbé and Isabella Lari

PhD in progress : Jérôme De Boeck, “Decomposition methods for combinatorial optimization problems", Octobre 2015, Bernard Fortz

PhD in progress : Léonard Von Niederhausern, "Approches bi-niveau pour la tarification de services énergéiques ”, Octobre 2015 , Luce Brotcorne