Improving the Neighborhood Selection Strategy in Simulated Annealing using the Optimal Stopping Problem

Simulated Annealing is one the most well known local search methods. In practice, it is often used to solve discrete optimization problems; especially very tough problems, [54, 37, 34, 3, 2, 41]. Global optimization is computationally extremely challenging and for large instances, exact methods reach their limitations quickly. Hence, in practice, often local optimization methods are used. Simulated annealing provides a powerful tool for escaping local optima by allowing moves to lower quality solutions with a pre-specified probability. Another big plus of Simulated Annealing is its ease of implementation. At each iteration of Simulated Annealing, the objective function value of the current solution and a new generated solution are compared. Improving moves are always accepted while only a fraction of non-improving moves are performed with the aim to escape local optimal solutions. The probability of accepting a non-improving move depends on the nonincreasing parameter of temperature. This technique comes from annealing in metallurgy. In this process, a metal is heated and slowly cooled off, in order to increase the size of its crystals while reducing the number of defects. The heating dissolves out atoms from their initial positions which can be seen as a local minimum with respect to energy level. Such atoms can then freely move around. Slowly cooling off has the effect that the free atoms can end up in positions with lower energy level than the initial positions. The crucial factor is to choose the cooling procedure appropriately, as cooling off to fast may not enable atoms to find better energy levels and cooling off to slowly is very time consuming. Simulated Annealing was introduced by Kirkpatrick et al. [33] in 1983 and independently by Černý [13] in 1985. It is an adaptation of a special Mont Carlo method generating sample states of a thermodynamic system which was introduced by Metropolis et al. in 1953, [38]. In 1986, Lundy and Mees were able to proof that under some technical assumptions Simulated Annealing converges with probability 1 to the global optimum, [36]. In this work we equip the simulated annealing algorithm with K neighborhood strategies and apply the Optimal Stopping Problem to determine the optimal time for changing the temperature. This study is organized as follows. In Section 2, we give an introduction to meta-heuristics in general and show the connection to Simulated Annealing. The concept of negative dynamic programing is introduced in Section 3. This provides the mathematical