Soft Variable Fixing in Path Relinking: An Application to ACO Codes

Soft variable fixing has emerged as one of the main techniques that the area of matheuristics can contribute to general metaheuristics. Recent years have in fact witnessed a fruitful interplay of methods that were originally proposed as general metaheuristcs with methods rooted in mathematic programming, which can be applied alone or as hybrids for solving combinatorial optimization problems. In this work, we show how one of the most effective matheuristics techniques, namely soft variable fixing, can be hybridized with Ant Colony Optimization. Specifically, we will combine a standard ACO code with a path relinking operator , implemented by means of soft variable fixing. Soft Variable Fixing is an operation based on Local Branching. This last is a technique originally introduced by Fischetti and Lodi [4], which works as follows. Starting from a feasible reference solution ¯ x of a mixed integer problem, the objective is to derive an exact exploration of a suitable neighborhood, defined on the binary representation of the reference solution. Following a positive integer parameter k, a k-OPT neighborhood N (¯ x, k) of ¯ x is defined as the set of the feasible solutions satisfying an additional local branching constraint. This is a constraint that counts the number of variables which change their value, and limits their number to be at most k. This permits to have the best solution which differs from ¯ x in at most k positions. Soft variable fixing has a wider scope than local branching, as it does not need to start with a full reference solution, but it can start with whichever subset of variables one need to fix and implement the local branching strategy in such a way as to ensure the feasibility of the optimized solutions. The overall idea of our work is to let the ACO explore the search space, keeping a pool of the best solutions encountered. Each time a new solution enters the pool, path relinking is performed toward other pool solutions. Path relinking is delegated to the use of a Mixed Integer Programming (MIP) solver. This can be actually done in different ways [3], here we instantiate a local branching [4] on a seed solution obtained by a combination of the decision variable values of the solutions we are relinking. As opposed to similar approaches, we allow to modify up to a bounded number of variables which are common to both endpoint solutions, hence …