On the Solution Existence of Nonconvex Quadratic Programming Problems in Hilbert Spaces
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.
KeywordsQuadratic 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
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.
- 5.Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer (2000)Google Scholar
- 11.Hauser, R.: The S-Procedure via dual cone calculus. arXiv:1305.2444 (2013)
- 14.Kurdila, A.J., Zabarankin, M.: Convex Functional Analysis. Birkhauser Verlag (2005)Google Scholar
- 17.Schochetman, I.E., Smith, R.L., Tsui, S.K.: Solution existence for infinite quadratic programming. Technical Report 97–10 (1997)Google Scholar
- 19.Rudin, W.: Functional Analysis. McGraw-Hill Inc (1991)Google Scholar