Disjunctive Inequalities: Applications and Extensions

where x ∈ R is the vector of decision variables, c ∈ R is a linear objective function and S ⊂ R is the set of feasible solutions of (1). Because S is generally hard to deal with, a possible approach for tackling (1) is to optimize the objective function over a suitable relaxation (i.e., easy to solve) P ⊇ S. Let x be the optimal solution over P . If x ∈ S the problem is solved. Otherwise, one can derive a valid inequality for S in order to separate x from S, i.e., an inequality αx ≥ β satisfied by all the feasible solutions in S and such that αx < β. The addition of the cutting plane αx ≥ β to the constraints defining P leads to a tighter relaxation P ′ = P ∩ {x ∈ R : αx ≥ β} and the process can eventually be iterated. The disjunctive approach to the separation problem, as introduced by Balas [5], considers defining an intermediate set Q ⊇ S not containing x and separating x from Q. The set Q is obtained by applying to P a valid disjunction D for the set S, such as