Abstract: The constrained discrete optimization (CDO) is transformed into a nonlinear constrained non-negative integer programming (CNIP) which can be solved by the proposed discrete differential evolution (DDE) algorithm that adopts several improvements such as the chaotic initialization of a population, the double-scheme mutation, and the integrating operator with stochastic perturbation. Aiming at the nonlinear constraints, the calculating approaches for the base penalty and the formula for the base function of continuous mapping are carried out, and self-adaptive penalty factors based on these notions for handling constraints are presented. It is studied that a novel measure, termed as a quasi re-averaging gene distance for a population, is employed to depict the diversity of the population and chaotic immigration operators depending on this measure and the probability are implemented to preserve the population diversity. Orientating constrained engineering optimizations, it proposes a novel hybrid DDE (HDDE) in which a self-adaptive method and a chaotic immigration strategy are dynamically incorporated in the DDE algorithm to improve its performance. Furthermore, three benchmark functions in CDO fields are utilized to test this HDDE algorithm and the results show that the new approach is robust and efficient, and is more optimal in objective functions than the discrete particle swarm optimization (DPSO) algorithm. Finally, this work also uses the proposed algorithm to optimize the transmission design of helical gear reducers and its objective value is better than those obtained by genetic algorithm (GA), improved GA (IGA), and DPSO, etc., and has decreased by 41% and 10% compared with GA and IGA, respectively.

Key words: Chaotic immigrant, Constrained discrete optimization, Differential evolution algorithm, Gene distance, Self-adaptive penalty function