A new restart procedure for combinatorial optimization and its convergence

TL;DR

This paper introduces a novel restart procedure for meta-heuristic algorithms in combinatorial optimization, demonstrating significant improvements in solution reliability and convergence speed, especially for large Traveling Salesman Problem instances.

Contribution

The paper presents a new iterative restart method based on failure probability surrogates, with proven convergence properties and practical effectiveness for large TSP problems.

Findings

Failure probability reduced by several orders of magnitude.

Significant gain in solution reliability over basic algorithms.

Convergence to optimal expected solution time proven theoretically.

Abstract

We propose a new iterative procedure to optimize the restart for meta-heuristic algorithms to solve combinatorial optimization, which uses independent algorithm executions. The new procedure consists of either adding new executions or extending along time the existing ones. This is done on the basis of a criterion that uses a surrogate of the algorithm failure probability, where the optimal solution is replaced by the best so far one. Therefore, it can be applied in practice. We prove that, with probability one, the restart time of the proposed procedure approaches, as the number of iterations diverges, the optimal value that minimizes the expected time to find the solution. We apply the proposed restart procedure to several Traveling Salesman Problem instances with hundreds or thousands of cities. As basic algorithm, we used different versions of an Ant Colony Optimization algorithm. We compare the results from the restart procedure with those from the basic algorithm. This is done by considering the failure probability of the two approaches for equal computation cost. This comparison showed a significant gain when applying the proposed restart procedure, whose failure probability is several orders of magnitude lower.