Multi-objective optimization of computationally expensive, multimodal problems is very challenging, and is even more difficult for problems with many objectives (more than three). Optimization methods that incorporate surrogates within iterative frameworks, can be effective for solving such problems by reducing the number of expensive objective function evaluations that need to be done to find a good solution. However, only a few surrogate algorithms have been developed that are suitable for solving expensive many-objective problems. We propose a novel and effective optimization algorithm, \(\varepsilon \)-MaSO, that integrates \(\varepsilon \)-dominance with iterative Radial Basis Function surrogate-assisted framework to solve problems with many expensive objectives. \(\varepsilon \)-MaSO also incorporates a new strategy for selecting points for expensive evaluations, that is specially designed for many-objective problems. Moreover, a bi-level restart mechanism is introduced to prevent the algorithm from remaining in a local optimum and hence, increase the probability of finding the global optimum. Effectiveness of \(\varepsilon \)-MaSO is illustrated via application to DTLZ test suite with 2 to 8 objectives and to a simulation model of an environmental application. Results on both test problems and the environmental application indicate that \(\varepsilon \)-MaSO outperforms the other two surrogate-assisted many-objective methods, CSEA and K-RVEA, and an evolutionary many-objective method Borg within limited budget.

计算昂贵的多模态问题的多目标优化非常具有挑战性，对于具有多个目标（超过三个）的问题则更加困难。在迭代框架中合并替代项的优化方法，可以通过减少寻找一个好的解决方案而需要进行的昂贵的目标函数评估的数量来有效解决此类问题。但是，仅开发了少数几种适合解决昂贵的多目标问题的代理算法。我们提出了一种新颖有效的优化算法\（\ varepsilon \）- MaSO，该算法将\（\ varepsilon \）- dominance与迭代径向基函数代理辅助框架进行了集成，以解决具有许多昂贵目标的问题。\（\ varepsilon \）- MaSO还采用了一种新的策略来为昂贵的评估选择点，该策略专门针对多目标问题而设计。此外，引入了双层重新启动机制以防止算法保留在局部最优中，因此增加了找到全局最优的可能性。的有效性\（\ varepsilon \） -MaSO经由应用到DTLZ测试套件被示为具有2至8个目标和环境应用的仿真模型。关于测试问题和环境应用的结果均表明，\（\ varepsilon \）- MaSO优于其他两种替代辅助的多目标方法CSEA和K-RVEA，以及在有限预算内进化的多目标方法Borg。