Qin Sitian, Le Xinyi, Wang Jun
IEEE Trans Neural Netw Learn Syst. 2017 Nov;28(11):2580-2591. doi: 10.1109/TNNLS.2016.2595489. Epub 2016 Aug 19.
This paper presents a neurodynamic optimization approach to bilevel quadratic programming (BQP). Based on the Karush-Kuhn-Tucker (KKT) theorem, the BQP problem is reduced to a one-level mathematical program subject to complementarity constraints (MPCC). It is proved that the global solution of the MPCC is the minimal one of the optimal solutions to multiple convex optimization subproblems. A recurrent neural network is developed for solving these convex optimization subproblems. From any initial state, the state of the proposed neural network is convergent to an equilibrium point of the neural network, which is just the optimal solution of the convex optimization subproblem. Compared with existing recurrent neural networks for BQP, the proposed neural network is guaranteed for delivering the exact optimal solutions to any convex BQP problems. Moreover, it is proved that the proposed neural network for bilevel linear programming is convergent to an equilibrium point in finite time. Finally, three numerical examples are elaborated to substantiate the efficacy of the proposed approach.
本文提出了一种用于双层二次规划（BQP）的神经动力学优化方法。基于卡鲁什 - 库恩 - 塔克（KKT）定理，将BQP问题简化为一个受互补约束的单层数学规划（MPCC）。证明了MPCC的全局解是多个凸优化子问题最优解中的最小解。开发了一种递归神经网络来求解这些凸优化子问题。从任何初始状态开始，所提出神经网络的状态收敛到神经网络的一个平衡点，该平衡点恰好是凸优化子问题的最优解。与现有的用于BQP的递归神经网络相比，所提出的神经网络能够保证为任何凸BQP问题提供精确的最优解。此外，证明了所提出的用于双层线性规划的神经网络在有限时间内收敛到一个平衡点。最后，详细阐述了三个数值例子以证实所提方法的有效性。
