Bsf-optimal Wya-optimal Schedule Schedule Number of Steps Bsf Quality Wya Quality Bsf Quality Wya Quality 5 2.038

12 schedules may be viewed as trading oo between reachability among solutions (high T) and a bias to lower-cost solutions (low T). The WYA criterion forces the low-cost bias to dominate at the end of the run, while the BSF criterion may allow reachability to take precedence. It is therefore not surprising that BSF-optimal schedules may actually be \warming" even at the end of the annealing execution (cf. our results for graph placment in 2]). In conclusion, the best-so-far analysis opens the door to new hill-climbing regimes, as well as new theoretical fronts such as the Markov analysis of best-so-far annealing. A major area of future research lies in the application of non-conventional temperature schedules to larger problem instances. To this end, our experiments point to periodic \iterated descent" methods 1] 8] 12] and adaptive construction of \non-monotone" (warming) schedules as being especially promising. We also believe that tuning BSF-optimal annealing strategies to the statistical parameters of optimization cost surfaces 17] will provide an important research direction. 6 Acknowledgements The authors wish to thank C.-W. Albert Tsao for his implementation of much of the code used in these experiments. References 1] E. B. Baum, Iterated descent: a better algorithm for local search in combinatorial optimization problems, Proc. Table 2: Expected BSF and WYA solution quality produced by locally optimal schedules for the 8-node graph bisection instance of Strenski and Kirkpatrick. We write the quality as a multiple of the cost of the global optimum solution. 5 Conclusions The far-reaching consequences of BSF analysis are apparent even at rst glance. For example, \innnite-time optimality" holds for a much wider range of schedules under the BSF criterion: given innnite time, any schedule that is bounded away from zero will eventually visit a globally optimal solution, assuming that all solutions are reachable at non-zero temperatures. In this paper, we have explored the implications of nite-time, best-so-far analysis of the simulated annealing algorithm. The BSF analysis is more consistent with annealing practice than the traditional WYA analysis, and the study of nite-time annealing also reeects practical reality, since applications of the annealing algorithm are certainly limited to nite amounts of CPU time. To assess the practical eeect of the BSF criterion on nite-time annealing strategies, we have numerically estimated BSF-and WYA-optimal schedules for small instances of two classic problem formulations. Our results show striking diierences between annealing schedules that optimize the two criteria. WYA-optimal …