## Section: Scientific Foundations

### Introduction

*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 Ilog-CPLEX
or FICO/Dash-Optimization's Xpress-mp). 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 the study of nonlinear models (non linearities are inherent in some engineering, economic and scientific applications) where solution techniques build on the best MIP approaches while demanding much more than simple extensions. Robust optimization is another area 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 mathematical programming (polyhedral approaches, Dantzig-Wolfe decomposition, non-linear integer programing, 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.