### Structure-driven fix-and-propagate heuristics for mixed integer programming

Gerald Gamrath, Timo Berthold, Stefan Heinz, Michael Winkler
2019 Mathematical Programming Computation
Primal heuristics play an important role in the solving of mixed integer programs (MIPs). They often provide good feasible solutions early in the solving process and help to solve instances to optimality faster. In this paper, we present a scheme for primal start heuristics that can be executed without previous knowledge of an LP solution or a previously found integer feasible solution. It uses global structures available within MIP solvers to iteratively fix integer variables and propagate
more » ... e fixings. Thereby, fixings are determined based on the predicted impact they have on the subsequent domain propagation. If sufficiently many variables can be fixed that way, the resulting problem is solved as an LP and the solution is rounded. If the rounded solution did not provide a feasible solution already, a sub-MIP is solved for the neighborhood defined by the variable fixings performed in the first phase. The global structures help to define a neighborhood that is with high probability significantly easier to process while (hopefully) still containing good feasible solutions. We present three primal heuristics that use this scheme based on different global structures. Our computational experiments on standard MIP test sets show that the proposed heuristics find solutions for about three out of five instances and therewith help to improve several performance measures for MIP solvers, including the primal integral and the average solving time. 1 of this paper, we denote by P(c, A, b, , u, N , I) a MIP of form (1) in dependence of the provided data. This modeling scheme allows to express many real-world optimization problems from various fields like production planning (Pochet and Wolsey, 2006) , scheduling (Heinz et al., 2013 ), transportation (Borndörfer et al., 2010, or telecommunication networks (Lee and Lewis, 2006) . On the other hand, the strict specifications for the problem statement make it possible to solve arising optimization problems for all these applications using the same algorithm. Therefore, very powerful generic solvers for MIPs have been developed over the last decades, which are used widely in research and practice Bixby, 2012; Achterberg and Wunderling, 2013) . These solvers are based on a branch-and-bound algorithm (Land and Doig, 1960; Dakin, 1965) , which is accelerated by various extensions. The basic concept is to split the problem into subproblems until they are easy enough to be solved. During this process, a lower bound is computed for each sub-problem by solving its linear programming (LP) relaxation P(c, A, b, , u, ∅), that is the problem obtained from (1) by omitting the integrality restrictions. At the same time, the objective value of the incumbent-the best feasible solution found so far-provides an upper bound on the global optimum. In combination, these bounds allow to speed up the solving process by disregarding sub-problems whose lower bound exceeds the upper bound since those cannot lead to an improving solution. It is evident that this algorithm profits directly from finding good solutions as early as possible. On the one hand, these solutions originate from integral LP relaxation solutions, on the other hand, so-called primal heuristics try to construct new feasible solutions or improve existing ones. Primal heuristics are incomplete methods without any success or quality guarantee which nevertheless are beneficial on average. There are different common approaches applied by many heuristics, e.g., rounding of the LP solution or diving, which iteratively changes the current sub-problem temporarily and solves the corresponding LP relaxation until an integral solution is obtained. For more details on primal heuristics, we refer to Berthold (2006) ; Fischetti and Lodi (2010); Berthold (2014a). In this paper, we introduce three novel heuristics which combine a fix-and-propagate scheme (Berthold and Gleixner, 2010; Achterberg and Wunderling, 2013) with the large neighborhood search (LNS) paradigm, cf. Danna et al. (2004) . The former is typically used for before-LP heuristics and iteratively fixes a variable and propagates this change to apply all implied changes to the domains of other variables. The latter defines a sub-problem, the neighborhood, by adding restrictions to the problem, and then solves this sub-problem as a MIP. A more detailed discussion of these heuristic concepts is given in Section 2. By modeling a specific problem as a MIP and solving it with a MIP solver, one profits from the decades of developments within this area. However, knowledge about the structure of the problem which could be exploited by a problem specific approach can hardly be fed into a MIP solver due to the generality of the algorithm. MIP solvers try to partially compensate this by detecting some common structures within the problem and exploiting them in the solving process. Examples for this are multi-commodity flow subproblems (Achterberg and Raack, 2010) and permutation structures (Salvagnin, 2014) . This detection is often done in the presolving phase, which is a preprocessing step trying to remove redundancies from the model and tighten the formulation. An overview of different global structures in MIP solvers and details about three of them, the clique table, the variable bound graph, and the variable locks, are given in Section 3. The heuristics presented in this paper use these global structures to apply a fix-and-propagate scheme. This means, they repeatedly fix variables and perform domain propagation to consider the direct consequences of these fixings on the domains of other variables. While this is a known approach in MIP heuristics Achterberg and Wunderling, 2013; Berthold and Hendel, 2015) , our new heuristics take a step further and make domain propagation their driving force.