Constraint Programming and Linear Programming for Quadratic 0-1 Problems
Résumé
In order to solve more easily combinatorial optimization problems, one way is to find theoretically a set of necessary or/and sufficient conditions
that the optimal solution of the problem has to verify. For instance, in linear programming, weak and strong duality conditions can be easily de-
rived. And in convex quadratic programming, Karush-Kuhn-Tucker conditions gives necessary and sufficient conditions. Despite that, in general,
such conditions doesn’t exist for integer programming, some necessary conditions can be derived from Karush-Kuhn-Tucker conditions for the
unconstrained quadratic 0-1 problem. In this paper, we present these conditions. We show how they can be used with constraint program-
ming techniques to fix directly some variables of the problem. Hence, reducing strongly the size of the problem. For example, for low density
problems of size lower than 50, those conditions combined with constraint programming may be sufficient to solve completely the problem, without
branching. In general, for quadratic 0-1 problems with linear constraints, we propose a new method combining these conditions with constraint and
linear programming. Some numerical results, with the instances of the OR-Library, will be presented.