An interactive approach based on a discrete differential evolution algorithm for a class of integer bilevel programming problems

Abstract

This paper presents an interactive approach based on a discrete differential evolution algorithm to solve a class of integer bilevel programming problems, in which integer decision variables are controlled by an upper-level decision maker and real-value or continuous decision variables are controlled by a lower-level decision maker. Using the Karush--Kuhn–Tucker optimality conditions in the lower-level programming, the original discrete bilevel formulation can be converted into a discrete single-level nonlinear programming problem with the complementarity constraints, and then the smoothing technique is applied to deal with the complementarity constraints. Finally, a discrete single-level nonlinear programming problem is obtained, and solved by an interactive approach. In each iteration, for each given upper-level discrete variable, a system of nonlinear equations including the lower-level variables and Lagrange multipliers is solved first, and then a discrete nonlinear programming problem only with inequality constraints is handled by using a discrete differential evolution algorithm. Simulation results show the effectiveness of the proposed approach.

Keywords:

• discrete bilevel programming problem
• smoothing function
• constraint-handling technique
• discrete differential evolution algorithm
• interactive approach

Acknowledgements

The authors wound like to thank the anonymous reviewers for their comments and suggestions for improving the article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Natural Science Basic Research Plan in Shaanxi Province of China [grant number 2013JM1022]; and Fundamental Research Funds for the Central Universities [grant number K50511700004]. The first author gratefully acknowledge the support from China Scholarship Council [grant number 201306965006].

Notes on contributors

Hong Li

Hong Li was born in 1972. He received his PhD degree in intelligent information processing from Xidian University in 2009. He is currently working as an associate professor in Xidian University. His research interests include evolutionary computation, intelligent information processing, optimisation theory, algorithms and applications.

Li Zhang

Li Zhang was born in 1976. She received the MS degree in applied mathematics and the PhD degree in signal and information processing both from Xidian University in 2003 and 2012. She is currently working as an associate professor in Xidian University. Her research interests include neural network, intelligent information processing.

Yong-Chang Jiao

Yong-Chang Jiao was born in 1964. He received his PhD degree from Xidian University in 1990. Currently, he is a professor with the Institute of Antennas and Electromagnetic Scattering, Xidian University. His research interests include optimisation algorithms and applications, evolutionary computation, as well as analysis and synthesis of antennas.