## Section: Scientific Foundations

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  ,  ,  . 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 (xi {0, 1} can be equivalently set as xi2 = xi ); sequencing constraints: for instance in scheduling problems, if job j follows job i , denoted by xij = 1 , then their completion times must satisfy Cj Ci + pj where pj is the processing time of j ; this can be modeled as xij (Cj-Ci-pj) 0 ; transitivity constraints represent requests of the type “if two variables xi and xj are equal to 1, a third one xk should be 1” (their quadratic formulation xk xixj is stronger than the linear xk xi + xj-1 when one relaxes the integrality restrictions).