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Saddle Point Criteria for Semi-infinite Programming Problems via an \(\eta \)-Approximation Method

  • Yadvendra Singh
  • S. K. Mishra
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 225)

Abstract

In this paper, we consider a semi-infinite programming problem involving differentiable invex functions. We construct an \(\eta \)-approximated semi-infinite programming problem associated with the original semi-infinite programming problem and establish relationship between its saddle point and an optimal solution. We also establish relationship between an optimal solution of original semi-infinite programming problem and saddle point of \(\eta \)-approximated semi-infinite programming problem. Examples are given to illustrate the obtained results.

Keywords

Semi-infinite programming Generalized convexity Optimality conditions 

Notes

Acknowledgements

The first author is supported by the Council of Scientific and Industrial Research(CSIR), New Delhi, India, through grant no. 09/013(0474)/2012-EMR-1.

References

  1. 1.
    Vaz, A.I.F., Fernandes, E.M.G.P., Gomes, M.P.S.F.: Robot trajectory planning with semi-infinite programming. Eur. J. Oper. Res. 153, 607–617 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Tong, X., Ling, C., Qi, L.: A semi-infinite programming algorithm for solving optimal power flow with transient stability constraints. J. Comput. Appl. Math. 217(2), 432–447 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Vaz, A.I.F., Ferreira, E.C.: Air pollution control with semi-infinite programming. Appl. Math. Model. 33(4), 1957–1969 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Winterfeld, A.: Application of general semi-infinite programming to lapidary cutting problems. Eur. J. Oper. Res. 191(3), 838–854 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    López, M.A., Vercher, E.: Optimality conditions for nondifferentiable convex semi-infinite programming. Math. Program. 27, 307–319 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kanzi, N.: Necessary optimality conditions for nonsmooth semi-infinite programming problems. J. Glob. Optim. 49, 713–725 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    López, M.A., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180, 491–518 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Shapiro, A.: Semi-infinite programming, duality, discretization and optimality conditions. Optimization 58, 133–161 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Antczak, T.: An \(\eta \)-approximation approach to nonlinear mathematical programming involving invex functions. Numer. Funct. Anal. Optim. 25, 423–438 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Antczak, T.: Saddle point criteria in an \(\eta \)-approximated method for nonlinear mathematical programming problem involving invex functions. J. Optim. Theory Appl. 132(1), 71–87 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ratiu, A., Duca, I.D.: Semi-infinite optimization problems and their approximations. Stud. Univ. Babes-Bolyai Math. 58(3), 401–411 (2013)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Mishra, S.K., Giorgi, G.: Invexity and Optimization. Springer, Berlin, Heidelberg (2008)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.C.M.P. Degree College (A Constituent Postgraduate College of Central University of Allahabad)AllahabadIndia
  2. 2.Department of MathematicsInstitute of Science, Banaras Hindu UniversityVaranasiIndia

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