Finite-Time Performance Analysis of Static Simulated Annealing Algorithms

Generalized hill climbing (GHC) algorithms provide a framework for modeling local search algorithms for addressing intractable discrete optimization problems. Current theoretical results are based on the assumption that the goal when addressing such problems is to find a globally optimal solution. However, from a practical point of view, solutions that are close enough to a globally optimal solution (where close enough is measured in terms of the objective function value) for a discrete optimization problem may be acceptable. This paper introduces β-acceptable solutions, where β is a value greater than or equal to the globally optimal objective function value. Moreover, measures for assessing the finite-time performance of GHC algorithms, in terms of identifying β-acceptable solutions, are defined. A variation of simulated annealing (SA), termed static simulated annealing (S2A), is analyzed using these measures. S2A uses a fixed cooling schedule during the algorithm's execution. Though S2A is provably nonconvergent, its finite-time performance can be assessed using the finite-time performance measures defined in terms of identifying β-acceptable solutions. Computational results with a randomly generated instance of the traveling salesman problem are reported to illustrate the results presented. These results show that upper and lower estimates for the number of iterations to reach a β-acceptable solution within a specified number of iterations can be obtained, and that these estimates are most accurate for moderate and high fixed temperature values for the S2A algorithm.