School of Computer Science and Technology, Qinghai Normal University, Xining, China.
School of Mathematics and Statistics, Qinghai Normal University, Xining, China.
PLoS One. 2020 Dec 17;15(12):e0243926. doi: 10.1371/journal.pone.0243926. eCollection 2020.
A bilevel programming problem with multiple objectives at the leader's and/or follower's levels, known as a bilevel multiobjective programming problem (BMPP), is extraordinarily hard as this problem accumulates the computational complexity of both hierarchical structures and multiobjective optimisation. As a strongly NP-hard problem, the BMPP incurs a significant computational cost in obtaining non-dominated solutions at both levels, and few studies have addressed this issue. In this study, an evolutionary algorithm is developed using surrogate optimisation models to solve such problems. First, a dynamic weighted sum method is adopted to address the follower's multiple objective cases, in which the follower's problem is categorised into several single-objective ones. Next, for each the leader's variable values, the optimal solutions to the transformed follower's programs can be approximated by adaptively improved surrogate models instead of solving the follower's problems. Finally, these techniques are embedded in MOEA/D, by which the leader's non-dominated solutions can be obtained. In addition, a heuristic crossover operator is designed using gradient information in the evolutionary procedure. The proposed algorithm is executed on some computational examples including linear and nonlinear cases, and the simulation results demonstrate the efficiency of the approach.
具有领导者和/或跟随者层面多个目标的双层规划问题，称为双层多目标规划问题（BMPP），由于该问题积累了分层结构和多目标优化的计算复杂性，因此极其困难。作为一个强 NP 难问题，BMPP 在获得两个层次上的非支配解时需要大量的计算成本，很少有研究解决这个问题。在这项研究中，使用代理优化模型开发了一种进化算法来解决此类问题。首先，采用动态加权和方法来解决跟随者的多目标情况，其中将跟随者的问题分为几个单目标问题。接下来，对于每个领导者变量值，都可以通过自适应改进的代理模型来近似最优解转换后的跟随者程序，而无需解决跟随者的问题。最后，将这些技术嵌入到 MOEA/D 中，从而可以获得领导者的非支配解。此外，在进化过程中使用梯度信息设计了启发式交叉算子。在所提出的算法中，在一些计算实例（包括线性和非线性情况）上进行了执行，仿真结果证明了该方法的有效性。
