Local search inequalities

We describe a general method for deriving new inequalities for integer programming formulations of combinatorial optimization problems. The inequalities, motivated by local search algorithms, are valid for all optimal solutions but not necessarily for all feasible solutions. These local search inequalities can help in either pruning the search tree at some nodes or in improving the bound of the LP relaxations. 1 Optimization and local search One of the most effective ways to solve an NP-hard combinatorial optimization problem is to formulate it as an integer program, which is in turn solved by branch and bound [9]. Let the formulation be zIP := min{c x : x ∈ S} (1) where S = {x : Ax ≥ b, x ≥ 0, x ∈ Z}. Moreover, let P = {x : Ax ≥ b, x ≥ 0} and P I = conv(S). Valid inequalities (also called cuts) can be added to (1) to strengthen the quality of the LP bound to be used in the branch-and-bound. The process of solving a combinatorial optimization problem with the addition of valid cuts is called branch-and-cut [11]. Local search is a general framework for finding good (not necessarily optimal) solutions of an optimization problem [1, 8]. In local search, a neighborhood function N (s) is specified, which, for a feasible solution s, defines a set of feasible solutions “close” to s. A local optimum is a solution s∗ such that c s∗ ≤ c s for all s ∈ N (s∗). Clearly, an optimal solution of the problem is also a local optimum for each possible neighborhood but not vice-versa. Local search heuristics usually work by quickly finding as many local optima as possible, and then returning the best one. Local search inequalities The results presented in this paper are based on the following observation: Given a local search neighborhood N , a global optimum of the problem must also be a local optimum for N . This is therefore an additional constraint on the global optimum. If it is possible to express the above constraint via linear inequalities, we call each such linear constraint a local search inequality (LSI). Basically, local search inequalities are constraints saying that, for each move which changes a feasible solution x into a feasible solution x′ ∈ N (x), it must be c x′ ≥ c x. These constraints are valid for local optima (and hence for global optima), but may be violated by some other feasible solutions in P I . Therefore, they cut through the set P I and are not valid inequalities in the usual sense.