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Accelerating the convergence of Genetic Algorithms (GAs) is a significant and promising research direction of evolutionary computation. In this paper, a novel Index-based GA (termed IndexGA) is proposed for the acceleration of convergence by reducing the number of fitness evaluations (FEs) in the reproduction procedure, i.e. the process of crossover and mutation. The algorithm divides the solution space into multiple regions, each represented by a unique index. Individuals in the IndexGA are redefined as indexes instead of solutions. In the reproduction procedure, an evaluated region is never evaluated again, and the fitness is directly obtained from the memory. Moreover, to improve the fitness of the promising regions, the algorithm performs an orthogonal local search (OLS) operator on the best-so-far region in each generation. Numerical experiments have been conducted on 13 benchmark functions and an application problem of power electronic circuit (PEC) to investigate the performance of IndexGA. The results show that the index-based strategy and the OLS in IndexGA significantly enhance the performance of GAs in terms of both convergence rate and solution accuracy.