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Abstract
In order to generate properly efficient solutions of a general multiobjective optimization problem, a unified direction approach was introduced by Ghaznavi et al. (Optim. Methods Softw., 2019). However, there are some deficiencies in this work, and a counterexample is given to show that properly efficient solutions of a general multiobjective optimization problem cannot be obtained via optimal solutions of the proposed scalarized problem. In the present paper, by adding slack variables and surplus variables, we modify the direction scheme proposed by Ghaznavi et al. (Optim. Methods Softw., 2019), and achieve necessary and sufficient conditions for (weakly and properly) efficient solutions of a general multiobjective optimization problem via optimal solutions of the corresponding scalarized problem. Further, the efficiency of our method is demonstrated by numerical examples. This work fills some gaps in the work of Ghaznavi et al. (Optim. Methods Softw., 2019).
Acknowledgments
The authors wish to thank the handling editor and anonymous references for their helpful comments and suggestions. Further, the authors are grateful to Professor R.S. Burachik and Associate Professor M.M. Rizvi for providing valuable suggestions on numerical experiments.
Disclosure statement
No potential conflict of interest was reported by the author(s).
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Notes on contributors
Liping Tang
Liping Tang received both the B.S. and M.S. degrees from Chongqing Normal University, Chongqing, China, in 2008 and 2011, respectively. She received the Ph.D. degree in operations research and cybernetics from Shanghai University, Shanghai, China, in 2015. She is currently an Associate Professor at College of Mathematics and Statistics, Chongqing Technology and Business University. Her current research interests include vector optimization, variational inequalities, set-valued analysis.
Xinmin Yang
Xinmin Yang received his Ph.D. degree in Applied Mathematics from Hong Kong Polytechnic University in 2002. At present, he is Professor and Head of National Center for Applied Mathematics of Chongqing, Chongqing Normal University, China, and he is also Director of the Key Laboratory for Optimization and Control, Ministry of Education, Chongqing Normal university, China. He was the recipient of the National Award for Natural Sciences of China in 2019. His research interests include machine learning, nonlinear programming.