Simulated Annealing Meta-heuristic for Addition Chain Optimization

Abstract

In this work, a simulated annealing (SA) algorithm is implemented in the Python programming language with the aim of minimizing addition chains of the "star-chain" type. The strategies for generating and mutating individuals are similar to those used by the evolutionary programming (EP) and genetic algorithms (GA) methods found in the literature [1]-[3]. The proposed variant is the acceptance mechanism that is based on the simulated annealing meta-heuristic (SA). The hypothesis is that with the proposed acceptance mechanism, diversity is obtained in the search-space through a simple strategy that allows finding better solutions compared to the deterministic method Optimized Window. The simulations were performed with exponents in the range 218-234 and were compared with the results reported in [3], where a GA is proposed to get optimal addition chains. It is concluded that the proposed algorithm is able to find chains of shorter length than those found with the Optimized Window method and with a performance similar to that of the GA proposed in [3].

An Evolutionary Algorithm for Short Addition Chains

“…On the other hand, many non-exact algorithms [4,[13][14][15][16]18,23,24,[26][27][28][29][30][31][32][33]36] have been proposed to find a short AC. In these algorithms, the length of the generated AC is not necessarily minimal and the algorithms run in polynomial time.…”

“…Also, the window and continued fraction methods give a better output than other methods. [13][14][15][16]23,[26][27][28][29][31][32][33] for a short AC include the genetic algorithm (GA), evolutionary algorithm (EA), ant colony algorithm, swarm intelligence algorithms, and artificial immune algorithm. All these algorithms are based on many factors such as the size of the population, the maximum number of iterations, and the strategies of different operators such as crossover and/or mutation.…”