On the Solution Existence of Nonconvex Quadratic Programming Problems in Hilbert Spaces

Abstract

We propose conditions for the existence of solutions for nonconvex quadratic programming problems whose constraint set is defined by finitely many convex linear-quadratic inequalities in Hilbert spaces. In order to obtain our results, we use either properties of the Legendre form or properties of compact operators with closed range. The results are established without requesting the convexity of the objective function or the compactness of the constraint set. As a special case, we obtain some on the existence of solutions results for the quadratic programming problems under linear constraints in Hilbert spaces.

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Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.39. The authors would like to thank Prof. N. D. Yen for valuable remarks and suggestions. The authors would like to express our sincere thanks to the anonymous referees and editors for insightful comments and useful suggestions.

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Correspondence to Vu Van Dong.

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Dong, V.V., Tam, N.N. On the Solution Existence of Nonconvex Quadratic Programming Problems in Hilbert Spaces. Acta Math Vietnam 43, 155–174 (2018). https://doi.org/10.1007/s40306-017-0237-9

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Keywords

  • Quadratic program in Hilbert spaces
  • Convex quadratic constraints
  • Solution existence
  • Legendre form
  • Recession cone
  • Compact operator with closed range

Mathematics Subject Classification (2010)

  • 90C20
  • 90C26
  • 90C30