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Polynomial Optimization on Some Unbounded Closed Semi-algebraic Sets


The article presents a study on a class of polynomial optimization problems over (noncompact) semi-algebraic sets which, by making changes of variables via suitable monomial mappings, become polynomial optimization problems over compact semi-algebraic feasible sets. It is known that the polynomial optimization problems on semi-algebraic feasible sets are satisfactory when the feasible sets are compact. Furthermore, determining whether a polynomial is bounded on such a semi-algebraic set can be replaced by checking whether its support lies in a closed and convex cone corresponding to the semi-algebraic set.

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This paper has been written while the second author is visiting Vietnam Institute for Advanced Study in Mathematics (VIASM), Hanoi, Vietnam. The author would like to thank the VIASM for hospitality and support. This research is partially funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED), Grant 101.04-2017.12 and by University of Transport and Communications under grant number T2019-CB-012. The authors would like to thank the referee for his valuable comments/suggestions.

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Correspondence to Toan M. Ho.

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Communicated by Vaithilingam Jeyakumar.

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Du, T.T., Ho, T.M. Polynomial Optimization on Some Unbounded Closed Semi-algebraic Sets. J Optim Theory Appl 183, 352–363 (2019).

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  • Sum of squares
  • Positivstellensatz
  • Polynomial optimization

Mathematics Subject Classification

  • 90C30
  • 14P10
  • 49K99