PUSH: a primal heuristic based on Feasibility PUmp and SHifting

Since MIP linear problems include both continuous and integer variables, they are proved to belong to the NP-hard class (see [38] for a more detailed analysis), meaning that they are not solvable in polynomial time. The complete exploration of the integer feasible set, whose cardinality grows exponentially with the number of variables, is yet possible to achieve the optimal solution, but for most of the practically significant instances, it would require unacceptable computational effort. In fact, the only way to solve to optimality any mixed-integer problem is to apply some of the well-known Branch and Bound techniques. However, despite combinatorial optimization community provided a great deal of these algorithms, for which the reader should refer to [31, 34, 16], MIP problems complexity is inherent with their belonging to NP-hard class. Therefore, when tackling MIP problems, one either seeks particular structures allowing to bring down the complexity, such as the availability, for a given class of problems, of the optimal formulation or exploits cutting plane generation to dramatically reduce the feasible region dimension. However, we often encounter MIP problems without having any prior knowledge of possible structures and, thus, pursuing the globally optimal solution could be in practice impossible or inefficient, since for our purpose a sub-optimal approximation is considered to be good enough. This makes heuristics one of the most widespread and feasible ways to achieve sub-optimal solutions of MIP problems within an affordable computational time. For the purpose of highlighting the perspective of our research, we can define two classes of MIP heuristics: improvement heuristics and start heuristics.