Constrained Extremum Problems, Regularity Conditions and Image Space Analysis. Part I: The Scalar Finite-Dimensional Case

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Abstract

Image space analysis has proved to be instrumental in unifying several theories, apparently disjoint from each other. With reference to constraint qualifications/regularity conditions in optimization, such an analysis has been recently introduced by Moldovan and Pellegrini. Based on this result, the present paper is a preliminary part of a work, which aims at exploiting the image space analysis to establish a general regularity condition for constrained extremum problems. The present part deals with scalar constrained extremum problems in a Euclidean space. The vector case as well as the case of infinite-dimensional image will be the subject of a subsequent part.

Keywords

Image space analysis Regularity condition Constraint qualification Separation function 

Mathematics Subject Classification

49N15 90C30 90C46 

Notes

Acknowledgements

The author is grateful to Professor Franco Giannessi and Professor Letizia Pellegrini for their constructive comments on this paper and their great help when this work was carried out during a stay of the author in the Department of Mathematics, University of Pisa. The author expresses his gratitude to the anonymous referees for their valuable comments and suggestions, which help to improve the paper. This research was supported by China Scholarship Council, the National Natural Science Foundation of China (Grants 11601437, 11526165) the Basic and Advanced Research of CQ CSTC (Grant cstc2015jcyja00038), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant KJ1501503) and the Chongqing Research Program of Basic Research and Frontier Technology (cstc2016jcyjA0270).

References

  1. 1.
    Giannessi, F.: Theorems of the alternative and optimality conditions. J. Optim. Theory Appl. 42, 331–365 (1984)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Karush, W.: Minima of functions of several variables with inequalities as side conditions. Master’s thesis, University of Chicago (1939)Google Scholar
  3. 3.
    John, F.: Extremum problems with inequalities as subsidiary conditions. In: Friedrichs, K.O., Neugebauer, O.E., Stoker, J.J. (eds.) Studies and Essays: Courant Anniversary Volume, pp. 187–204. Interscience, New York (1948)Google Scholar
  4. 4.
    Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Neyman, J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 481–492. University of California Press, Berkeley (1951)Google Scholar
  5. 5.
    Peterson, D.W.: A review of constraint qualifications in finite-dimensional spaces. SIAM Rev. 15, 639–654 (1973)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Eustaquio, R.G., Karas, E.W., Ribeiro, A.A.: Constraint qualification for nonlinear programming. Technical report, Federal University of Parana (2010)Google Scholar
  7. 7.
    Flores-Bazán, F., Mastroeni, G.: Strong duality in cone constrained nonconvex optimization. SIAM J. Optim. 23, 153–169 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Flores-Bazán, F., Mastroeni, G.: Characterizing FJ and KKT conditions in nonconvex mathematical programming with applications. SIAM J. Optim. 25, 647–676 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. Part I: sufficient optimality conditions. J. Optim. Theory Appl. 142, 147–163 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. Part II: necessary optimality conditions. J. Optim. Theory Appl. 142, 165–183 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Giannessi, F.: Constrained Optimization and Image Space Analysis: Separation of Sets and Optimality Conditions, vol. 1. Springer, New York (2005)MATHGoogle Scholar
  12. 12.
    Quang, P.H., Yen, N.D.: New proof of a theorem of F. Giannessi. J. Optim. Theory Appl. 68, 385–387 (1991)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Zhu, S.K., Li, S.J.: Unified duality theory for constrained extremum problems. Part I: image space analysis. J. Optim. Theory Appl. 161, 738–762 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Zhu, S.K., Li, S.J.: Unified duality theory for constrained extremum problems. Part II: special duality schemes. J. Optim. Theory Appl. 161, 763–782 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Giannessi, F., Mastroeni, G.: Separation of sets and Wolfe duality. J. Glob. Optim. 42, 401–412 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Dien, P.H., Mastroeni, G., Pappalardo, M., Quang, P.H.: Regularity conditions for constrained extremum problems via image space. J. Optim. Theory Appl. 80, 19–37 (1994)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Rubinov, A.M., Uderzo, A.: On global optimality conditions via separation functions. J. Optim. Theory Appl. 109, 345–370 (2001)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Rubinov, A.M., Yang, X.Q.: Lagrange-Type Functions in Constrained Non-convex Optimization. Kluwer Academic Publishers, Dordrecht (2003)CrossRefMATHGoogle Scholar
  19. 19.
    Giannessi, F.: Some perspectives on vector optimization via image space analysis. J. Optim. Theory Appl. 177(3) (2018)Google Scholar
  20. 20.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  21. 21.
    Luo, H.Z., Mastroeni, G., Wu, H.X.: Separation approach for augmented Lagrangians in constrained nonconvex optimization. J. Optim. Theory Appl. 144, 275–290 (2010)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Gould, F.J., Howe, S.: A new result on interpreting Lagrange multipliers as dual variables. Technical report no. 738, The Institute of Statistics, University of North Carolina (1971)Google Scholar
  23. 23.
    Flegel, M. L.: Constraint qualifications and stationarity concepts for mathematical programs with equilibrium constraints. Ph.D. thesis, Institute of Applied Mathematics and Statistics, University of Würzburg (2005)Google Scholar
  24. 24.
    Guo, L., Lin, G.H., Ye, J.J.: Stability analysis for parametric mathematical programs with geometric constraints and its applications. SIAM J. Optim. 22, 1151–1176 (2012)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Economics and MathematicsSouthwestern University of Finance and EconomicsChengduChina
  2. 2.Department of MathematicsUniversity of PisaPisaItaly

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