Notes on the Optimization Problems Corresponding to Polynomial Complementarity Problems

  • Vu Trung HieuEmail author
  • Yimin Wei
  • Jen-Chih Yao


This work is motivated by a conjecture of Che et al. (J Optim Theory Appl 168:475–487, 2016) which says that if the feasible region of a tensor complementarity problem is nonempty, then the corresponding optimization problem has a solution. The aim of the paper is twofold. First, we show several sufficient conditions for the solution existence of the optimization problems corresponding to polynomial complementarity problems. Consequently, some results for tensor complementarity problems are obtained. Second, we disprove the conjecture by giving a counterexample.


Polynomial complementarity problem Tensor complementarity problem Polynomial optimization problem Feasible region Solution existence 

Mathematics Subject Classification

90C33 11C08 



The authors are grateful to the editor-in-chief and the anonymous referees for their valuable suggestions. The first author wishes to thank the Department of Applied Mathematics, National Sun Yat-Sen University and the Center for General Education, China Medical University for hospitality and support. The research of Yimin Wei is supported by the National Natural Science Foundation of China under Grant 11771099 and the Innovation Program of Shanghai Municipal Education Committee.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Division of MathematicsPhuong Dong UniversityHanoiVietnam
  2. 2.School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied MathematicsFudan UniversityShanghaiPeople’s Republic of China
  3. 3.Center for General EducationChina Medical UniversityTaichungTaiwan

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