Abstract
Local search is a fundamental tool in the development of heuristic algorithms. A neighborhood operator takes a current solution and returns a set of similar solutions, denoted as neighbors. In best improvement local search, the best of the neighboring solutions replaces the current solution in each iteration. On the other hand, in first improvement local search, the neighborhood is only explored until any improving solution is found, which then replaces the current solution. In this work we propose a new strategy for local search that attempts to avoid low-quality local optima by selecting in each iteration the improving neighbor that has the fewest possible attributes in common with local optima. To this end, it uses inequalities previously used as optimality cuts in the context of integer linear programming. The novel method, referred to as delayed improvement local search, is implemented and evaluated using the travelling salesman problem with the 2-opt neighborhood and the max-cut problem with the 1-flip neighborhood as test cases. Computational results show that the new strategy, while slower, obtains better local optima compared to the traditional local search strategies. The comparison is favourable to the new strategy in experiments with fixed computation time or with a fixed target.
Highlights
Finding an optimal solution to a given instance of a combinatorial optimization problem can be a hard computational task
Most of the methods used to solve hard combinatorial optimization problems fall into two major groups: exact methods, whose goal is to find an optimal solution of the problem together with a proof of optimality, and heuristic methods, which seek to find a sufficiently good solution within a reasonable time limit
The main objective of this article is the proposal of a new strategy for local search procedures applied to combinatorial optimization problems
Summary
Finding an optimal solution to a given instance of a combinatorial optimization problem can be a hard computational task. Local search procedures start from a feasible initial solution and proceed by applying operations on the current solution to obtain better solutions. The set of solutions that can be generated with a given operator on a solution s is called the neighborhood of s. From the neighborhood of the current solution, one feasible solution that improves the value of the objective function. A solution having no improving solution in its neighborhood is called a local optimum. Let N be a neighborhood operator that from a given solution produces a set of modified solutions. A solution returned by the local search is a local optimum of the considered neighborhood
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