## Section: Scientific Foundations

### Quadratic approaches

The communication between the non-linear programming and the
combinatorial optimization communities is limited although the
latter has much to learn from the former. Several of the latest
developments in discrete optimization are imports from convex
optimization [59] , [89] , [66] . Quadratic
programming (QP), in particular, offers very powerful modeling
tools. A *quadratic program * is a formulation in continuous
variables whose cost and/or constraint functions are polynomials of
degree 2. Several classical constraint types for combinatorial
problems are more efficiently modeled with (QP): binary constraints
(x_{i}{0, 1} can be equivalently set as x_{i}^{2} = x_{i} );
sequencing constraints: for instance in scheduling problems, if job
j follows job i , denoted by x_{ij} = 1 , then their completion
times must satisfy C_{j}C_{i} + p_{j} where p_{j} is the
processing time of j ; this can be modeled as x_{ij} (C_{j}-C_{i}-p_{j})0 ; transitivity constraints represent requests of the
type “if two variables x_{i} and x_{j} are equal to 1, a third
one x_{k} should be 1” (their quadratic formulation x_{k}x_{i}x_{j} is stronger than the linear x_{k}x_{i} + x_{j}-1 when one
relaxes the integrality restrictions).

Although Quadratic Programming (QP) is NP-Hard, some special cases
or relaxations are polynomially solvable. Minimizing a convex
quadratic cost over a feasible region described by linear
constraints is easy. This method is implemented in commercial MIP
solvers. Convex QP with linear constraints and 0-1 variables can
then be solved using branch-and-bound and solving the continuous QP
relaxation at each node. Also, the *Semi-Definite Programming*
(SDP) relaxation of a QP is polynomially solvable. An SDP is an
extension of an LP where variables are the components of a matrix
that is constrained to be semi-definite. When the objective is not
convex, it can be convexified using an augmented relaxation approach
[60] , [59] : the Hessian is made convex by adding
to the objectives a weighted sum of the quadratic binary constraints
and the squared norm of equality constraints (optimized weights are
obtained by solving an SDP relaxation). When applied to an already
convex objective, this approach is still useful to improve the
continuous relaxation bound. Solvers are available for SDP
[55] , but, in their current implementation, they are
very sensitive to the conditioning of the
matrices.

Even though the numerical solution of QP and SDP remains problematic, recent applications of these techniques to some combinatorial problems have led to major improvements [117] , [107] , [90] , [59] . Generally, an SDP relaxation can be found for any general Integer Program [142] . Starting from the QP reformulation of an IP, a Lagrangian relaxation procedure is applied to yield an SDP: after Lagrangian dualization, the (unconstrained) QP has a solution (in the primal variables) iff some matrix of coefficients is positive semidefinite. The associated SDP bound is always better than classical LP relaxation [113] , although sometimes the size of the SDP formulation is problematic. In former studies, we discovered other SDP formulations for Lovász's bound on vertex coloring [119] : a direct quadratization that appears to be intermediate between the ones of [107] and [117] . The SDP formulation obtained by application of the general scheme of [142] is of huge dimension and, because of symmetry, does not bring more than our (compact) SDP formulation. Yet, it can be fruitfully used to compute bounds on generalizations of Vertex Coloring where symmetry does not hold (List Coloring, some problems of Frequency Assignment) [119] .