Towards a Lower Bound for the Average Case Runtime of Simulated Annealing on TSP

TL;DR

This paper provides theoretical bounds on the expected performance of simulated annealing for the Traveling Salesperson Problem, showing it often remains far from optimal unless run for exponentially many iterations.

Contribution

It establishes lower bounds on SA's solution quality and variance, extending analysis to nonstationary cases, and introduces a conjecture about SA's convergence behavior.

Findings

SA explores solutions far from the global optimum under optimal cooling schedules

Bounds on the expected tour length and variance are derived for random TSP instances

Numerical evidence suggests SA requires exponential time to find near-optimal solutions

Abstract

We analyze simulated annealing (SA) for simple randomized instances of the Traveling Salesperson Problem. Our analysis shows that the theoretically optimal cooling schedule of Hajek explores members of the solution set which are in expectation far from the global optimum. We obtain a lower bound on the expected length of the final tour obtained by SA on these random instances. In addition, we also obtain an upper bound on the expected value of its variance. These bounds assume that the Markov chain that describes SA is stationary, a situation that does not truly hold in practice. Hence, we also formulate conditions under which the bounds extend to the nonstationary case. These bounds are obtained by comparing the tour length distribution to a related distribution. We furthermore provide numerical evidence for a stochastic dominance relation that appears to exist between these two distributions, and formulate a conjecture in this direction. If proved, this conjecture implies that SA stays far from the global optimum with high probability when executed for any sub-exponential number of iterations. This would show that SA requires at least exponentially many iterations to reach a global optimum with nonvanishing probability.