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Separations and Optimality of Constrained Multiobjective Optimization via Improvement Sets

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Abstract

In this paper, we investigate the separations and optimality conditions for the optimal solution defined by the improvement set of a constrained multiobjective optimization problem. We introduce a vector-valued regular weak separation function and a scalar weak separation function via a nonlinear scalarization function defined in terms of an improvement set. The nonlinear separation between the image of the multiobjective optimization problem and an improvement set in the image space is established by the scalar weak separation function. Saddle point type optimality conditions for the optimal solution of the multiobjective optimization problem are established, respectively, by the nonlinear and linear separation methods. We also obtain the relationships between the optimal solution and approximate efficient solution of the multiobjective optimization problem. Finally, sufficient and necessary conditions for the (regular) linear separation between the approximate image of the multiobjective optimization problem and a convex cone are also presented.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions, which have helped to improve the paper. This research was partially supported by the Natural Science Foundation of China (11401487, 11571055), the Basic and Advanced Research Project of Chongqing (cstc2016jcyjA0239), the China Postdoctoral Science Foundation (2015M582512), the Fundamental Research Funds for the Central Universities (XDJK2018).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Jiawei Chen & La Huang

  2. College of Computer Science, Chongqing University, Chongqing, 400044, China

    Jiawei Chen

  3. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Shengjie Li

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  1. Jiawei Chen
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  2. La Huang
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  3. Shengjie Li
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Correspondence to Shengjie Li.

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Chen, J., Huang, L. & Li, S. Separations and Optimality of Constrained Multiobjective Optimization via Improvement Sets. J Optim Theory Appl 178, 794–823 (2018). https://doi.org/10.1007/s10957-018-1325-2

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