Abstract
Considering that average convergence rate estimation of clonal selection algorithms is a difficult problem and is still in its infancy, this article researches the convergence rate of an elitist clonal selection algorithm. It derives the best individual transition probability matrix from the directional transition probability of best individuals in algorithm populations and constructs matrix norms that meet certain requirements to resolve difficulties in calculating the matrix caused by large algorithm populations in practical applications, thereby proposing a simple and effective method of estimating average convergence rate of the algorithm. In addition, simulation experiments are performed to validate universality and validity of the estimation method.
Highlights
Artificial immune algorithm is an intelligent algorithm that is inspired by biological immune mechanisms and utilized to solve complex issues
Scholars have made fruitful achievements in studying clonal selection algorithms in recent years, the studies focus on algorithm implementation, improvement, and engineering application, and the theoretical analyses concentrate on algorithm convergence
Convergence rate, which is an important part of theoretical researches on clonal selection algorithms, illustrates the algorithm convergence and plays an important role in establishing proper stropping criteria and measurement standards to comprehensively and objectively evaluate execution strategies.[3]
Summary
Artificial immune algorithm is an intelligent algorithm that is inspired by biological immune mechanisms and utilized to solve complex issues. Due to the lack of effective analytical methods, there are few reports on convergence rate of clonal selection algorithms and no strict theoretical conclusions. Clonal selection is one of the most famous biological immunological mechanisms.[9] Clonal selection algorithm, which is based on clonal selection, takes unknown objective functions as the antigens and possible optimal solutions as the antibodies. It is possible to estimate the convergence rate of elitist clonal selection algorithms by virtue of the best individual transition probability matrix. The state transition matrix of clonal selection algorithms is a random matrix (a homogeneous Markov chain) and the matrix Q is a constant random matrix. State transition of the best individuals constitutes a limited and homogeneous Markov chain and state 1 is the only absorption state of the Markov chain On this basis, Theorem 1 and Lemma 1 are given.
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