Best-so-far vs. where-you-are: implications for optimal finite-time annealing

Abstract

The simulated annealing (SA) algorithm is widely used for heuristic global optimization due to its high-quality results and its ability, in theory, to yield optimal solutions with probability one. Standard SA implementations use monotone decreasing, or ‘cooling’ temperature schedules that are motivated by the algorithm's proof of optimality as well as by an analogy with statistical thermodynamics. In this paper, we challenge this motivation. The theoretical framework under which monotone cooling schedules are ‘optimal’ fails to capture the practical performance of the algorithm; we therefore propose a ‘best-so-far⌉ (BSF) criterion that measures the practical utility of a given annealing schedule. For small instances of two classic combinatorial problems, we determine annealing schedules that are optimal in terms of expected cost of the output solution. When the goal is to optimized the cost of the last solution seen by the algorithm (the ‘where-you-are⌉ (WYA) criterion used in previous theoretical analyses), we confirm the traditional wisdom of cooling temperature schedules. However, if the goal is to optimize the cost of the best solution seen over the entire algorithm execution (i.e., the BSF criterion), we give evidence that optimal schedules do not decrease monotonically toward zero, and are in fact periodic or warming. These results open up many interesting research issues, including the BSF analysis of simulated annealing and how to best apply hill-climbing to difficult global optimizations.