Symbolic regression is to search the space of mathematical expressions to find a model that best fits a given dataset. As genetic programming (GP) with the tree representation can represent solutions as expression trees, it is popularly-used for regression. However, GP tends to evolve unnecessarily large programs (known as bloat), causing excessive use of CPU time/memory and evolving solutions with poor generalization ability. Moreover, even though the importance of local search has been proved in augmenting the search ability of GP (termed as memetic algorithms), local search is underused in GP-based methods. This work aims to handle the above problems simultaneously. To control bloat, a multi-objective (MO) technique (NSGA-II, Non-dominant Sorting Genetic Algorithm) is selected to incorporate with GP, forming a multi-objective GP (MOGP). Moreover, three mutation-based local search operators are designed and incorporated with MOGP respectively to form three multi-objective memetic algorithms (MOMA), i.e. MOMA_MR (MOMA with Mutation-based Random search), MOMA_MF (MOMA with Mutation-based Function search) and MOMA_MC (MOMA with Mutation-based Constant search). The proposed methods are tested on both benchmark functions and real-world applications, and are compared with both GP-based (i.e. GP and MOGP) and nonGP-based symbolic regression methods. Compared with GP-based methods, the proposed methods can reduce the risk of bloat with the evolved solutions significantly smaller than GP solutions, and the local search strategies introduced in the proposed methods can improve their search ability with the evolved solutions dominating MOGP solutions. In addition, among the three proposed methods, MOMA_MR performs best in RMSE for testing, yet it consumes more training time than others. Moreover, compared with six reference nonGP-based symbolic regression methods, MOMA_MR generally performs better than or similar to them consistently.

符号回归是搜索数学表达式的空间，以找到最适合给定数据集的模型。由于具有树表示的遗传编程（GP）可以将解决方案表示为表达树，因此它广泛用于回归分析。但是，GP趋向于不必要地扩展大型程序（称为膨胀程序），从而导致CPU时间/内存的过度使用以及泛化能力差的解决方案的发展。此外，尽管已经证明了局部搜索在增强GP（称为模因算法）的搜索能力方面的重要性，但基于GP的方法并未充分利用局部搜索。这项工作旨在同时解决上述问题。为了控制膨胀，选择了多目标（MO）技术（NSGA-II，非优势排序遗传算法）以与GP结合使用，从而形成多目标GP（MOGP）。此外，设计了三个基于突变的局部搜索算子，并将其分别与MOGP结合，以形成三个多目标模因算法（MOMA），即MOMA_MR（具有基于突变的随机搜索的MOMA），MOMA_MF（具有基于突变的功能搜索的MOMA）和MOMA_MC（带有基于突变的常量搜索的MOMA）。所提出的方法在基准功能和实际应用中均经过测试，并与基于GP的符号回归方法（即GP和MOGP）和基于非GP的符号回归方法进行了比较。与基于GP的方法相比，所提出的方法可以显着地减小进化解决方案的风险，而与GP解决方案相比，该方法可以降低膨胀的风险，并且所提出的方法中引入的局部搜索策略可以通过以MOGP解决方案为主导的进化解决方案提高其搜索能力。此外，在这三种建议的方法中，MOMA_MR在RMSE的测试中表现最好，但比其他方法要花费更多的训练时间。此外，与六种基于非GP的参考符号回归方法相比，MOMA_MR的性能通常要好于或始终如一。