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Solving implicit mathematical programs with fuzzy variational inequality constraints based on the method of centres with entropic regularization

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Abstract

The purpose of this paper is to consider a class of mathematical programs with fuzzy implicit variational inequality constraints in finite dimension real spaces. By using the “tolerance approach” and the fuzzy set theory, we also show that solving the fuzzy mathematical program problem with fuzzy implicit variational inequality constraints is equivalent to solving a fuzzy implicit complementarity constrained optimization problem, and the fuzzy implicit complementarity constrained optimization problem can be converted to a regular nonlinear parametric programming problem. Further, a new smoothing approach based on a version of the “method of centres” with entropic regularization for solving the resulting optimization problem and our main results are presented and a numerical example is provided to illustrate our main results applying quasi-Newton line search of MATLAB software.

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Acknowledgments

This work was supported by Sichuan Province Cultivation Fund Project of Academic and Technical Leaders, and the Open Research Fund of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things (2013WZJ01), and has been partially supported by Ministerio de Economia y Competitividad (Spain), Project MTM2010-15314, and cofinanced by the European Community fund FEDER. We are grateful to the reviewers for their helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, People’s Republic of China

    Heng-you Lan

  2. Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782, Santiago de Compostela, Spain

    Juan J. Nieto

  3. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Juan J. Nieto

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  1. Heng-you Lan
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  2. Juan J. Nieto
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Correspondence to Heng-you Lan.

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Lan, Hy., Nieto, J.J. Solving implicit mathematical programs with fuzzy variational inequality constraints based on the method of centres with entropic regularization. Fuzzy Optim Decis Making 14, 493–511 (2015). https://doi.org/10.1007/s10700-015-9207-7

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