## Section: Overall Objectives

### Overall Objectives

Quantitative modeling is routinely used in both
industry and administration to design and operate transportation,
distribution, or production systems. Optimization concerns every
stage of the decision-making process: long
term investment budgeting and activity planning, tactical management of scarce resources, or the control
of day-to-day operations. In many optimization problems that arise in
decision support applications the most important decisions (control
variables) are discrete in
nature: such as
on/off decision to buy, to invest, to hire, to send a vehicle, to allocate
resources, to decide on precedence in operation
planning, or to install a connection in network design.
Such *combinatorial optimization* problems can be modeled as
linear or nonlinear
programs with integer decision variables and extra variables to
deal with continuous adjustments. The most widely used modeling
tool consists in defining the feasible decisions set using linear inequalities with a mix of integer and
continuous variables, so-called Mixed Integer Programs (MIP), which already allow a fair description of
reality and are
also well-suited for global optimization. The solution of such models is essentially based on enumeration
techniques and is notoriously difficult given the huge size of the
solution space. Commercial solvers have made significant progress
but remain quickly overwhelmed beyond a certain problem size. A key
to further progress is the development of better problem
formulations that provide strong continuous approximations and hence
help to prune the enumerative solution scheme. Effective solution
schemes are a complex blend of techniques: cutting planes to better
approximate the convex hull of feasible (integer) solutions,
Lagrangian decomposition methods to produce powerful relaxations,
constraint programming to actively reduce the solution domain through
logical implications, heuristics and meta-heuristics (greedy, local improvement, or randomized partial search
procedures) to produce good candidate solutions,
and branch-and-bound or dynamic programming enumeration schemes to find a global optimum. The
real challenge is to integrate
the most efficient methods in one global system so as to prune what is
essentially an enumeration based solution technique.

Building on complementary expertise, our team's overall goals are threefold:

- Objective $\left(i\right)$
To design tight formulations for specific problems and generic models, relying on delayed cut and column generation, decomposition, extended formulations and projection tools for linear and nonlinear mixed integer programming models. More broadly, to contribute to theoretical and methodological developments of exact approaches in combinatorial optimization, while extending the scope of applications (in particular to encompass nonlinear models).

- Objective $\left(ii\right)$
To demonstrate the strength of cooperation between complementary exact mathematical optimization techniques, constraint programming, combinatorial algorithms and graph theory. To develop “efficient” algorithms for specific mathematical models and to tackle large-scale real-life applications, providing provably good approximate solutions by combining exact methods and heuristics.

- Objective $\left(iii\right)$
To provide prototypes of specific model solvers and generic software tools that build on our research developments, writing proof-of-concept code, while making our research findings available to internal and external users.