Mixed-Integer Programming Approaches for Risk-Averse Multicriteria Optimization

风险规避优化旨在解决存在不确定性的决策问题，这些不确定性涉及低概率但后果严重的事件。此外，决策者在面对这些问题时往往需要考虑多个相互冲突的性能标准。尽管它们无处不在，影响广泛，但涉及多个标准、离散决策和追索行动的风险规避优化问题尚未得到很好的研究。该项目的目标是通过开发新的模型和有效的方法来弥补这一差距，这些模型和方法将增强我们的知识库，并使解决更现实和更普遍的风险规避多标准优化问题成为可能。特别要研究的问题是在国土安全和人道主义救济应用方面普遍存在的问题。通过开发解决这些问题的模型和有效方法，该项目将提高我们准备和应对国家紧急情况的能力。该项目涉及三个主要重点，将推动规避风险的多准则优化模型和方法的发展。首先，在单阶段优化领域，提出了一种新的风险规避多目标优化模型，其中多个标准的相对重要性是模糊的。为此，必须定义一种新的具有理想理论性质的鲁棒多变量风险度量。在本项目中，将这种风险度量的优化问题表述为凹最小化问题，并将开发有效的求解方法。其次，对多变量风险下优化的切面生成问题中出现的基本非凸多面体子结构进行严格的研究。第三，制定多元风险下的两阶段优化问题，以便进行追索权决策。对于这些问题，将设计涉及第二阶段问题的连续线性近似的分解算法。

Risk-averse optimization aims to address decision-making problems in the presence of uncertainty that involves events of low probability but severe consequences. In addition, decision makers are often required to consider multiple and conflicting performance criteria when faced with such problems. Despite their ubiquity and wide-ranging impact, risk-averse optimization problems that involve multiple criteria, discrete decisions and recourse actions are not well studied. The goal of this project is to bridge this gap by developing novel models and effective methods that will enhance our knowledge base, and enable the solution of more realistic and general risk-averse multicriteria optimization problems. In particular, the problems that will be studied are prevalent in homeland security and humanitarian relief applications. By developing models and effective methods for such problems, this project will improve our ability to prepare for and respond to national emergencies.This project involves three major thrusts that will advance the development of models and methods for risk-averse multicriteria optimization. First, in the realm of single-stage optimization, a novel model for risk-averse multiobjective optimization is proposed, where the relative importance of the multiple criteria is ambiguous. To this end, a new robust multivariate risk measure with desirable theoretical properties must be defined. In this project, the problem of optimizing such a risk measure will be formulated as a concave minimization problem for which effective solution methods will be developed. Second, a rigorous study of the fundamental non-convex polyhedral substructures arising in the cut generation problems for optimization under multivariate risk will be conducted. Third, two-stage optimization problems under multivariate risk will be formulated to allow recourse decisions. For these problems, decomposition algorithms that involve successive linear approximations of the second-stage problems will be devised.