[论文解读] Multicriteria Optimization and Decision Making: Principles, Algorithms and Case Studies

Michael Emmerich, André Deutz|arXiv (Cornell University)|Jun 29, 2024

Advanced Multi-Objective Optimization Algorithms参考文献 90被引用 8

一句话总结

本工作为多目标优化与决策制定（MODA）提供了广泛的入门介绍，涵盖形式定义、帕累托概念、标量化，以及具有教学强调和案例研究的代表性求解方法。

ABSTRACT

Real-world decision and optimization problems, often involve constraints and conflicting criteria. For example, choosing a travel method must balance speed, cost, environmental footprint, and convenience. Similarly, designing an industrial process must consider safety, environmental impact, and cost efficiency. Ideal solutions where all objectives are optimally met are rare; instead, we seek good compromises and aim to avoid lose-lose scenarios. Multicriteria optimization offers computational techniques to compute Pareto optimal solutions, aiding decision analysis and decision making. This reader offers an introduction to this topic and has been developed on the basis of the revised edition of the reader for the MSc computer science course "Multicriteria Optimization and Decision Analysis" at the Leiden Institute of Advanced Computer Science, Leiden University, The Netherlands. This course was taught annually by the first author from 2007 to 2023 as a single semester course with lectures and practicals. Our aim was to make the material accessible to MSc students who do not study mathematics as their core discipline by introducing basic numerical analysis concepts when necessary and providing numerical examples for interesting cases. The introduction is organized in a unique didactic manner developed by the authors, starting from more simple concepts such as linear programming and single-point methods, and advancing from these to more difficult concepts such as optimality conditions for nonlinear optimization and set-oriented solution algorithms. Besides, we focus on the mathematical modeling and foundations rather than on specific algorithms, though not excluding the discussion of some representative examples of solution algorithms.

研究动机与目标

• 将多目标优化与决策制定作为一个系统设计问题进行动机说明和定义。
• 引入形式化的问题定义、序关系和帕累托概念，为MODA奠定基础。
• 介绍核心求解方法，包括标量化、帕累托前沿方法，以及转化为单目标问题。
• 讨论景观分析、最优性条件，以及算法方法（确定性与进化性）。
• 提供一个以教学为导向、以示例为驱动的介绍，面向硕士研究生和新接触MCO/MCDM数学的读者。

提出的方法

• 使用标准的数学规划记号對多目标优化进行形式化。
• 描述帕累托支配、序关系以及帕累托前沿的几何性质。
• 解释标量化方法（线性与非线性聚合、多属性效用理论、到参考点的距离）及其与单目标重构的关系。
• 介绍将多准则问题转化为带约束的单目标问题（ε-约束/折衷规划）。
• 概述确定性帕累托前沿计算和进化多目标优化（策略/支配与多样性、基于指标的方法）。
• 提供以案例研究驱动的说明，强调基础而非特定算法。

实验结果

研究问题

• RQ1 多目标优化问题如何在数学规划框架下形式化定义和分类？
• RQ2 支配与帕累托有效性的哪些概念构成多目标问题解集的结构？
• RQ3 多目标如何转化为单目标形式，以及对解的质量有哪些影响？
• RQ4 近似帕累托前沿的核心算法族有哪些，包括确定性与进化方法？
• RQ5 哪些基础概念将优化理论与现实世界中的决策相连接？

主要发现

• 笔记提供从简单到高级概念的结构化路径，将优化理论与决策分析联系起来。
• 它们定义理解MCO/MCDM所必需的形式问题类别和帕累托概念。
• 它们将标量化技术与帕累托前沿方法联系起来，并讨论单目标重构。
• 它们展示了帕累托优化的确定性方法与进化方法，强调支配与多样性之间的权衡。
• 它们强调在MODA中决策辅助工具的作用以及以人为本的考量。
• 它们提供案例研究视角，阐明在经济、工程、医学与社会科学中的应用。