Many real world optimization problems are formulated as mixed-variable optimization problems (MVOPs) which involve both continuous and discrete variables. MVOPs including dimensional variables are characterized by a variable-size search space. Depending on the values of dimensional variables, the number and type of the variables of the problem can vary dynamically. MVOPs and variable-size MVOPs (VMVOPs) are difficult to solve and raise a number of scientific challenges in the design of metaheuristics. Standard metaheuristics have been first designed to address continuous or discrete optimization problems, and are not able to tackle (V)MVOPs in an efficient way. The development of metaheuristics for solving such problems has attracted the attention of many researchers and is increasingly popular. However, to our knowledge there is no well established taxonomy and comprehensive survey for handling this important family of optimization problems. This paper presents a unified taxonomy for metaheuristic solutions for solving (V)MVOPs in an attempt to provide a common terminology and classification mechanisms. It provides a general mathematical formulation and concepts of (V)MVOPs, and identifies the various solving methodologies than can be applied in metaheuristics. The advantages, the weaknesses and the limitations of the presented methodologies are discussed. The proposed taxonomy also allows to identify some open research issues which needs further in-depth investigations.

翻译：许多实际优化问题被建模为混合变量优化问题（MVOPs），这类问题同时包含连续变量和离散变量。包含维度变量的MVOPs具有可变尺寸搜索空间的特征，问题的变量数量与类型会随维度变量的取值动态变化。MVOPs与变尺寸MVOPs（VMVOPs）的求解难度较大，在元启发式算法的设计层面引发了多项科学挑战。标准元启发式算法最初是为解决连续或离散优化问题而设计的，无法高效处理（变尺寸）MVOPs。开发针对这类问题的元启发式方法已吸引众多研究者的关注，并呈现日益增长的研究趋势。然而，据我们所知，目前尚缺乏针对这一重要优化问题族群的成熟分类体系与全面综述。本文提出面向（变尺寸）MVOPs的元启发式求解统一分类框架，旨在建立通用术语与分类机制。文章给出了（变尺寸）MVOPs的一般数学形式化定义及核心概念，系统梳理了可应用于元启发式算法的各类求解方法论，并深入讨论了这些方法的优势、缺陷与局限性。所提出的分类框架还识别了若干亟待深入探索的开放性研究问题。