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_i² = 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_jC_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_kx_ix_j is stronger than the linear x_kx_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] .