Genetic programming (GP) approaches are among the state-of-the-art for symbolic regression, the task of constructing symbolic expressions that fit well with data. To find highly accurate symbolic expressions, both the expression structure and any contained real-valued constants, are important. GP-GOMEA, a modern model-based evolutionary algorithm, is one of the leading algorithms for finding accurate, yet compact expressions. Yet, GP-GOMEA does not perform dedicated constant optimization, but rather uses ephemeral random constants. Hence, the accuracy of GP-GOMEA may well still be improved upon by the incorporation of a constant optimization mechanism. Existing research into mixed discrete-continuous optimization with EAs has shown that a simultaneous and well-integrated approach to optimizing both discrete and continuous parts, leads to the best results on a variety of problems, especially when there are interactions between these parts. In this paper, we therefore propose a novel approach where constants in expressions are optimized at the same time as the expression structure by merging the real-valued variant of GOMEA with GP-GOMEA. The proposed approach is compared to other forms of handling constants in GP-GOMEA, and in the context of other commonly used techniques such as linear scaling, restarts, and constant tuning after GP optimization. Our results indicate that our novel approach generally performs best and confirms the importance of simultaneous constant optimization during evolution.

翻译：遗传编程（GP）方法是符号回归任务（即构建与数据良好拟合的符号表达式）的先进技术之一。为获得高精度的符号表达式，表达式结构及其包含的实值常量均至关重要。GP-GOMEA作为一种基于模型的现代进化算法，是寻找精确且紧凑表达式的主流算法之一。然而，GP-GOMEA并未进行专门的常量优化，而是采用瞬时随机常量。因此，引入常量优化机制有望进一步提升GP-GOMEA的精度。现有关于进化算法中离散-连续混合优化的研究表明，对离散部分与连续部分进行同步且高度集成的优化，能在多种问题上取得最优结果，特别是在这两部分存在交互作用时。基于此，本文提出一种新方法，通过将GOMEA的实值变体与GP-GOMEA融合，在优化表达式结构的同时同步优化表达式中的常量。我们将所提方法与GP-GOMEA中其他常量处理方式（如线性缩放、重启及GP优化后常量调优等常用技术）进行了比较。结果表明，我们提出的新方法整体表现最佳，证实了演化过程中同步常量优化的重要性。