双层优化问题的某些机理分析及相关求解策略研究

At present, the bilevel programming problem is transformed into a single level programming problem by virtue of the lower level optimal value reformulation or the Karush-Kuhn-Tucker (KKT) one. Particularly, it can be transformed into a mathematical programming problem with equilibrium/complementarity constraints (MPECs/MPCCs, for short) via the Karush-Kuhn-Tucker (KKT) conditions of the lower level problem if the lower problem is convex and the Slater constraint qualification is satisfied. However, how to transform the bilevel programming into a single level one when the lower problem is convex but the Slater constraint qualification does not hold or the lower problem is nonconvex? The purpose of the project is to discuss the other related theoretical analysis and solution methods of the bilevel optimization problem. The main contents are as follows: (1) Based on the application background and related theory of bilevel optimization problems, some mechanism analysis is discussed. (2) When the lower problem is nonconvex or the Slater constraint qualification does not hold, the related theoretical analysis and solution methods of this bilevel optimization problem are given. (3) New intuitionistic fuzzy interactive methods are established for solving the hierarchical level programming problem. This method can not only embody the subjective intention and objective environment of the decision maker, but also reflect the personalization of decision makers to the choice of solutions. Therefore, the upper and lower policymakers can obtain the optimal strategy to win-win. (4) We build new principal-agent models and electricity market models and design the optimal contract to achieve the goal of making each decision-maker win. (5) Numerical and intelligent optimization algorithms are constructed to solve the bilevel optimization problem effectively, and numerical examples and experiments are given to illustrate the rationality of the model and the effectiveness of the algorithm.

目前，求解双层规划一般方法是下层最优值函数技术和KKT转化策略。当下层凸且满足Slater约束规格时，可由KKT转化策略将其转化为一MPECs\MPCCs问题。然而下层凸但不满足Slater约束规格或下层非凸时，如何将其转化为单层规划有待研究。本项目旨在探讨双层优化问题的相关理论及其求解策略，主要研究内容为：(1)根据双层优化应用背景及相关理论，探讨双层优化问题的某些机理分析；(2)探讨下层非凸或不满足Slater约束规格的双层优化问题的相关理论与算法等问题；(3)运用直觉模糊理论设计新的直觉模糊交互式方法，求得既能体现决策者主观意识且符合客观环境又能反映决策者对解选择的个性化特征的满意解，使上下层决策者获得共赢的策略；(4)构建新的委托代理和电力市场模型，为各决策者提供有效激励机制，达到共赢的目标；(5)构造有效的数值算法和智能算法，利用数值实验和实例分析说明模型的合理性和算法的有效性。

双层优化问题具有主从递阶决策的特征，上下层决策者的满意策略至关重要，从双层优化问题的自身特点及其应用背景出发，探讨其相关的机理特征及其理论具有一定理论意义和应用价值。本项目主要研究内容及成果为：.(1)双层优化问题的相关理论与算法。针对黎曼流形上双层优化问题、双层多目标规划问题、单领导多随从问题、多领导随从问题和线性双层规划问题，得到了其最优性条件并构造有效的数值优化算法和智能算法。尤其是针对下层为凸但不满足Slater约束规格的双层规划问题，通过对下层问题进行扰动，得到了原双层规划问题的近似解；针对不同决策者的个性选择，定义新的得分函数，结合效用函数及直觉模糊理论构造了相关的决策模型，提出了一种新的直接模糊交互式方法。.(2)互补约束优化问题的研究。基于罚函数法构造了二阶锥混合互补问题的广义低阶罚算法；利用函数投影到二阶锥上的光滑近似提出了二阶锥线性互补问题的近似低阶惩罚方法；将摄动问题的中心路径重新表述为等效方程组，每次迭代只使用全牛顿步长提出了加权线性互补问题的一种全修正牛顿不可行内点方法。.(3)开关约束优化问题的研究。引进新的约束规格并给出了强、弱、M、B稳定点的最优性必要条件；给出了开关约束问题的Wofle型对偶模型并讨论了它的弱、强、逆、受限逆和严格逆对偶结果；构造了有效的增广拉格朗日求解算法。.(4)其他优化问题的研究。如不确定数据优化问题的鲁棒有效解、集值优化的最优性条件、稀疏优化的算法设计和多目标优化与变分不等式理论研究等等。