## Section: Research Program

### Introduction

integer programming, graph theory, decomposition approaches, polyhedral approaches, quadratic programming approaches, constraint programming.

*Combinatorial optimization* is the field of discrete optimization problems. In many
applications, the most important decisions (control variables) are binary (on/off decisions)
or integer (indivisible quantities). Extra variables can represent continuous adjustments or
amounts. This results in models known as *mixed integer programs* (MIP), where the
relationships between variables and input parameters are expressed as linear constraints and
the goal is defined as a linear objective function. MIPs are notoriously difficult to solve:
good quality estimations of the optimal value (bounds) are required to prune
enumeration-based global-optimization algorithms whose complexity is exponential. In the
standard approach to solving an MIP is so-called *branch-and-bound algorithm* : $\left(i\right)$ one
solves the linear programming (LP) relaxation using the simplex method; $\left(ii\right)$ if the LP
solution is not integer, one adds a disjunctive constraint on a factional component (rounding
it up or down) that defines two sub-problems; $\left(iii\right)$ one applies this procedure recursively,
thus defining a binary enumeration tree that can be pruned by comparing the local LP bound to
the best known integer solution. Commercial MIP solvers are essentially based on
branch-and-bound (such IBM-CPLEX, FICO-Xpress-mp, or GUROBI). They have made tremendous
progress over the last decade (with a speedup by a factor of 60). But extending their
capabilities remains a continuous challenge; given the combinatorial explosion inherent to
enumerative solution techniques, they remain quickly overwhelmed beyond a certain problem
size or complexity.

Progress can be expected from the development of tighter formulations. Central to our field is the characterization of polyhedra defining or approximating the solution set and combinatorial algorithms to identify “efficiently” a minimum cost solution or separate an unfeasible point. With properly chosen formulations, exact optimization tools can be competitive with other methods (such as meta-heuristics) in constructing good approximate solutions within limited computational time, and of course has the important advantage of being able to provide a performance guarantee through the relaxation bounds. Decomposition techniques are implicitly leading to better problem formulation as well, while constraint propagation are tools from artificial intelligence to further improve formulation through intensive preprocessing. A new trend is robust optimization where recent progress have been made: the aim is to produce optimized solutions that remain of good quality even if the problem data has stochastic variations. In all cases, the study of specific models and challenging industrial applications is quite relevant because developments made into a specific context can become generic tools over time and see their way into commercial software.

Our project brings together researchers with expertise in mathematical programming (polyhedral approaches, decomposition and reformulation techniques in mixed integer programing, robust and stochastic programming, and dynamic programming), graph theory (characterization of graph properties, combinatorial algorithms) and constraint programming in the aim of producing better quality formulations and developing new methods to exploit these formulations. These new results are then applied to find high quality solutions for practical combinatorial problems such as routing, network design, planning, scheduling, cutting and packing problems, High Performance and Cloud Computing.