基于值函数内点罚方法的双层规划问题研究

Research on Bi-Level Programming Problem Based on Value-Function-Based Interior-Point Penalty Method

本文考虑了一个在实际中有很多应用的双层规划问题,其下层问题带有多个不等式约束。为了开发有效的数值算法,通常需要将双层规划转化为单层优化问题。本文首先提出了罚函数,然后利用罚函数将下层问题的约束函数罚到目标函数得到逼近下层问题的无约束优化问题,并证明该无约束优化问题的最优值函数逼近于原下层问题的最优值函数,从而利用该逼近最优值函数将双层规划问题近似为一系列单层优化问题,并证明该一系列单层优化问题的解收敛于原双层规划的最优解。

In this paper, we consider a bi-level programming with many applications in practice, where the lower-level problem comes with multiple inequality constraints. In order to develop efficient nu-merical algorithms, it is often necessary to transform the bi-level programming into a single-level optimization problem. In this paper, we first propose a penalty function, and then use the penalty function to penalize the constraint function of the lower-level problem to the objective function to obtain an unconstrained optimization problem that approximates the lower-level problem. And we prove that the optimal value function of this unconstrained optimization problem approximates the optimal value function of the original lower-level problem. The bi-level programming problem is approximated to a series of single-level optimization problems by using the approximate optimal value function, and the solutions of the series of single-level optimization problems are proved to converge to the optimal solutions of the original bi-level programming.