Average Convergence Rate of Evolutionary Algorithms II: Continuous Optimization

A good convergence metric must satisfy two requirements: feasible in calculation and rigorous in analysis. The average convergence rate is proposed as a new measurement for evaluating the convergence speed of evolutionary algorithms over consecutive generations. Its calculation is simple in practice and it is applicable to both continuous and discrete optimization. Previously a theoretical study of the average convergence rate was conducted for discrete optimization. This paper makes a further analysis for continuous optimization. First, the strategies of generating new solutions are classified into two categories: landscape-invariant and landscape-adaptive. Then, it is proven that the average convergence rate of evolutionary algorithms using landscape-invariant generators converges to zero, while the rate of algorithms using positive-adaptive generators has a positive limit. Finally, two case studies, the minimization problems of the two-dimensional sphere function and Rastrigin function, are made for demonstrating the applicability of the theory.