A Permutation-Translation Simulated Annealing Algorithm for L1 and L2 Unidimensional Scaling

A latent class MDS model with spatial constraints for non-stationary spatial covariance estimation

“…Furthermore, in all the test runs described in the present paper, the given solution produced high attraction rates, i.e., the percentage of times that the lowest optimum value was found during the 20 replications, an indicator of the algorithm's efficiency in terms of solution quality, as defined in Murillo et al (2005) and in Vera et al (2007a). Thus, the problem of local minima inherent to the EM algorithm is less severe when Simulated Annealing is used.…”

“…Then, the cooling factor c controlling the rate of temperature decrease, and the truncating length of the Markov chain at each temperature level LC, which is increased by a fixed number IC every m iterations, are initialized. The final system temperature T f is chosen to be very close to zero, ensuring that the algorithm ends at least in a local optimum, and the initial temperature T 0 is calculated following a random sample procedure adopted in previous implementations, for example in Murillo et al (2005), or in Vera et al (2007a). A probability value of v, and a maximum of Ma max random allocations are set to average Ma possible increases of the solution that worsen the conditional log-likelihood, to obtain an initial temperature such that in the first iterations 100v% of the worst solutions are accepted.…”

Non-stationary spatial covariance structure estimation in oversampled domains by cluster differences scaling with spatial constraints

“…Therefore, an extension with an exchange method of reallocation based on simulated annealing in a cooling schedule similar to that of Murillo et al (2005) or Vera et al (2007) could be more appropriate. Nevertheless, the consideration of spatial contiguity constraints involves one cycle of length usually less than N, and in all test runs in the present paper the problem of empty classes did not appear.…”

The Local Minima Problem in Hierarchical Classes Analysis: An Evaluation of a Simulated Annealing Algorithm and Various Multistart Procedures

“…Simulated annealing (SA), based on an analogy to a metallurgical cooling process, is a local search technique that is often used to solve problems of combinatorial data analysis such as Kmeans clustering (see, e.g., Al-Sultan & Khan, 1996;Klein & Dubes, 1989) and unidimensional scaling (see, e.g., Brusco & Stahl, 2000;Murillo et al, 2005). Given a possible solution to a combinatorial optimization problem-the current solution-an SA algorithm generates a new solution-the trial solution-by randomly changing one or more parameter values of the current solution.…”

“…Not surprisingly, a great deal of research effort has therefore been put into the search for strategies for avoiding them. In particular, many authors advocate the following two strategies: using local search techniques such as simulated annealing, genetic algorithms, or tabu search (see, e.g., Al-Sultan & Khan, 1996;Brusco, 2001;Murillo, Vera, & Heiser, 2005) and/or implementing multistart procedures with a number of good starting values or a very large number of random starting values (see, e.g., Hand & Krzanowski, 2005;Milligan, 1980;Steinley, 2003). In this paper we will therefore investigate to which extent the local minima problem in hierarchical classes analysis can be mitigated by means of simulated annealing (SA) and/or by using various multistart procedures.…”