弱线性双层规划问题的理论与算法研究

A bilevel programming problem is a class of hierarchical nesting structure of the nonconvex optimization problem. A weak bilevel programming is a special bilevel programming in which the upper level decision maker is risk-averse, and it plays exceedingly important role in different application fields, such as production planning, principal-agent, toll pricing and others. Our project aims to study the theories and algorithms of weak linear bilevel programming. The main research contents are summarized as follows. (1) For a weak linear bilevel programming with a single follower, we will characterize the relevant geometric properties. On the other hand, we will propose a reduced method. (2) For a weak linear bilevel programming with multiple followers, based on the actual circumstances that the followers whether exchange information or reference the partial decision-making information, this project will construct two common weak linear bilevel multi-follower programming: the weak linear bilevel multi-follower programming with independent followers and the weak linear bilevel multi-follower programming in a referential-uncooperative situation. Furthermore, we will explore their fundamental properties, design the solution algorithms, analyze the actual case studies, illustrate the rationality of the proposed models and the feasibility of the developed algorithms. Finally, we will check the obtained optimal solution results are of importance to the leader who makes an effective decision-making. The successful implementation of this project not only will have a positive impact on the study of nonlinear programming problems, but also will further promote the application of weak linear bilevel programming.

双层规划是一类具有主从递阶嵌套的非凸优化问题。弱双层规划是指上层决策者具有风险规避的一类特殊双层规划，它在生产计划、委托代理以及收费定价等领域有着广泛而重要的应用。本项目旨在探讨弱线性双层规划的理论和算法，主要研究内容如下：(1) 针对单随从弱线性双层规划，一是刻画有关几何性质，二是设计一个简约求解方法；(2) 针对多随从弱线性双层规划，基于各随从之间有无信息交流、是否相互交流参考部分决策信息这些实际情况，拟构建两类常见的多随从弱线性双层规划(独立多随从弱线性双层规划和参考-非合作的多随从弱线性双层规划)模型，探讨其基本性质，设计求解算法，分析实际案例，阐明模型的合理性与算法的可行性，验证所获得的最优解结果对上层决策者制定有效的决策有着重要的参考价值。该项目的成功实施将对非线性规划问题的研究产生积极影响，还将进一步推动弱线性双层规划的应用研究。

本项目研究了弱线性双层规划问题的理论、算法及其应用。具体内容如下：. (1) 弱线性双层规划问题的研究。详细分析了弱线性双层规划问题的结构，探讨了它的有关几何性质，设计了Kth-Best算法、简约方法、罚函数方法、基于分枝定界的算法以及全局优化算法。. (2) 多随从的弱线性双层规划问题的研究。首先从实际的生产分销计划问题出发，构建了独立多随从的弱线性双层规划的数学模型，借助罚函数方法和KKT最优性条件，将此类问题转化为一个传统的单层优化问题，得到了一些有价值的结果。此外，还设计了部分共享多随从的弱线性双层规划问题的数学模型，给出了相关最优解的定义，设计了一种求解算法。. (3) 其他双层规划问题的研究。针对下层问题具有有限个最优解的一类悲观双层规划问题，提出了一种极大熵方法；针对多层规划问题，设计了三种直觉模糊交互式算法；综述了弱双层规划问题的解定义、解的存在性、最优性条件、求解复杂性、求解方法和应用。. 截至目前，已在包括Information Sciences、Optimization等国内外期刊发表10篇论文，研究成果丰富和完善了数学规划的理论、算法和应用。