双层规划相关模型的理论、算法及应用

The bilevel program is widely used in the fields of management, engineering, etc.. At present, more results are about the bilevel program with a convex lower level problem, while the bilevel program with a nonconvex problem or a vector optimization problem are difficult to study and the results are rare. This project will study three problems: 1. Stability analysis of semivector bilevel program. The bilevel program with a vector optimization lower level problem is called the semivector bilevel programming model. We will firstly consider the lower semi-continuity of the various effective solution mapping of the lower level vector optimization. Then we discuss the existence of the solution of the bilevel programming problem and the continuity of the optimal value function based on this. 2. Piecewise convexification method for solving bilevel program with a nonconvex lower level problem. We will replace the lower level nonconvex objective function with piecewise convex function, then the original problem reformulated into a mathematical program with equilibrium constrained (MPEC). Finally, we discuss the relationship between various stationary points and local optimal solution of MPEC and those of original bilevel program. 3. Semivector bilevel programming model for real-time electricity pricing of smart grids. In order to improve the efficiency of power utilization and the safety of power supply, more attention needs to the real-time electricity pricing of the grid.We will establish a semivector bilevel programming model for real-time electricity pricing of smart grid, and discuss the relationship between the four types of solutions (pessimistic, optimistic, rewarding, and deceiving) and the solving algorithms.

双层规划广泛应用于经管、工程等领域。目前研究较多的是下层为凸问题的情形，而对于下层为非凸问题或向量优化问题的情形，由于其研究难度较大成果相对较少。本项目将研究三方面的问题：1.半向量双层规划模型的稳定性。下层为向量优化问题的双层规划称为半向量双层规划。我们将研究参数向量优化的多类有效解集映射的下半连续性，并基于此讨论半向量双层规划解的存在性和最值函数的连续性。2.下层为非凸问题的双层规划模型的逐段凸化算法。我们将用凸函数逐段近似替代下层的非凸目标函数，从而将原问题近似转化为均衡约束规划问题求解，并讨论转化问题的各类稳定点和最优解与原问题的稳定点和最优解的关系。3.智能电网实时定价的半向量双层规划模型。为提高电力能源利用效率和供应安全性需对智能电网的实时定价投入更多的关注。我们将建立智能电网实时定价的半向量双层规划模型，并讨论该模型的四类解(悲观解、乐观解、奖励解、惩罚解)的关系及求解方法。

本项目主要研究了双层规划及其相关问题的理论、算法及应用。最主要的研究内容有如下四项：1.提出了一个求解下层非凸的双层规划问题的逐段凸化算法。通过BB方法逐段凸化下层问题，从而将原问题近似转化均衡约束规划问题，并讨论了均衡约束规划问题的全局/局部最优解与原问题的全局/局部最优解之间的关系。我们的方法是当前求解该类问题最有效的方法之一，可广泛应用于经济、管理等领域。2.讨论了半向量双层规划问题的稳定性。我们首先给出了参数多目标规划问题有效解集映射和弱有效解集映射的下半连续性的条件，然后将该条件应用于半向量双层规划问题得到其最优值函数和最优解集映射的连续性定理。该项成果可为双层规划和多目标规划的算法构造和收敛性分析提供理论支持。3.首次研究了混合整数半无限规划问题。我们从生产计划安排，城市固体垃圾管理问题中首次抽象出混合整数半无限规划模型，然后将其等价转化为均衡约束规划或者非线性规划问题，并构造了行生产算法求解该问题。我们构造的算法可以求解大规模混合整数半无限规划问题。4.提出了存零约束优化问题的增广Lagrangian方法, 在MPSC-RCPLD 条件假设下证明了算法生成的解序列的收敛性。该算法可用于求解证券投资交易中的either-or约束优化模型。