Mixed-Integer Programming Approaches for Risk-Averse Multicriteria Optimization

风险厌恶优化的目标是在存在不确定性的情况下解决决策问题，这些不确定性涉及到概率较低但后果严重的事件。此外，决策者在面对这类问题时，往往需要考虑多个相互冲突的业绩标准。尽管风险厌恶优化问题无处不在，影响广泛，但涉及多目标、离散决策和追索权行动的风险厌恶优化问题还没有得到很好的研究。这个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，这些新模型和有效方法将增强我们的知识库，并使更现实和更普遍的风险规避多准则优化问题的解决成为可能。特别是，将研究的问题普遍存在于国土安全和人道主义救济应用中。通过为此类问题开发模型和有效方法，该项目将提高我们应对和应对国家突发事件的能力。该项目涉及三个主要推动力，将推动风险厌恶多目标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，其中多个目标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稳健多元风险度量。在这个项目中，优化风险度量的问题将被描述为一个凹极小化问题，并将为其开发有效的求解方法。其次，对多元风险优化割集生成问题中出现的基本非凸多面体子结构进行了严格的研究。第三，多变量风险下的两阶段优化问题将被描述为允许追索权决策。对于这些问题，将设计涉及第二阶段问题的逐次线性逼近的分解算法。

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.