On First-Order Conditions for Optimality of Nondifferentiable Semi-infinite Programming

  • Ali SadeghiehEmail author
Research Paper


The purpose of this paper is to give some new Karush–Kuhn–Tucker-type necessary optimality conditions for nonsmooth semi-infinite problems. Moreover, we present some suitable examples for our results. The paper is organized by Fréchet and Mordukhovich subdifferentials.


Semi-infinite optimization Constraint qualification Optimality condition Subdifferential 


  1. Clarke FM (1983) Optimization and nonsmooth analysis. Wiley, HobokenzbMATHGoogle Scholar
  2. Deepmala (2014) A study on fixed point theorems for nonlinear contractions and its applications. Ph.D. Thesis, Pt. Ravishankar Shukla University, RaipurGoogle Scholar
  3. Deepmala Mishra VN, Marasi HR, Shabanian H, Nosraty M (2017) Solution of Voltra-Fredholm Integro-Differential equations using Chebyshev collocation method. Global J Technol Optim 8:210. Google Scholar
  4. Fajardo MD, López MA (1999) Locally Farkas-Minkowski systems in convex semi-infinite programming. J Optim Theory Appl 103:313–335MathSciNetCrossRefzbMATHGoogle Scholar
  5. Goberna MA, López MA (1998) Linear semi-infinite optimization. Wiley, HobokenzbMATHGoogle Scholar
  6. Golestani M, Kanzi N (2017) Necessary and sufficient conditions for optimality of nonsmooth semi-infinite programming. Iran J Sci Technol Trans Sci 41:923–929MathSciNetCrossRefGoogle Scholar
  7. Hettich R, Kortanek O (1993) Semi-infinite programming: theory, methods, and applications. SIAM Rev 35:380–429MathSciNetCrossRefzbMATHGoogle Scholar
  8. Hiriart-Urruty HB, Lemarechal C (1991) Convex analysis and minimization algorithms. I & II. Springer, BerlinzbMATHGoogle Scholar
  9. Kanzi N (2011) Necessary optimality conditions for nonsmooth semi-infinite programming Problems. J Global Optim 49:713–725MathSciNetCrossRefzbMATHGoogle Scholar
  10. Kanzi N (2014) Constraint qualifications in semi-infinite systems and their applications in nonsmooth semi-infinite problems with mixed constraints. SIAM J Optim 24:559–572MathSciNetCrossRefzbMATHGoogle Scholar
  11. Kanzi N (2014) Two constraint qualifications for non-differentiable semi-infinite programming problems using Fréchet and Mordukhovich subdifferentials. J Math Ext 8:83–94MathSciNetzbMATHGoogle Scholar
  12. Kanzi N (2015) On strong KKT optimality conditions for multiobjective semi-infinite programming problems with lipschitzian data. Optim Lett 9:1121–1129MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kanzi N, Nobakhtian S (2008) Optimality conditions for nonsmooth semi-infinite programming. Optimization 59:717–727CrossRefzbMATHGoogle Scholar
  14. Kanzi N, Nobakhtian S (2009) Nonsmooth semi-infinite programming problems with mixed constraints. J Math Anal Appl 351:170–181MathSciNetCrossRefzbMATHGoogle Scholar
  15. Li W, Nahak C, Singer I (2000) Constraint qualifications in semi-infinite system of convex inequalities. SIAM J Optim 11:31–52MathSciNetCrossRefzbMATHGoogle Scholar
  16. López MA, Still G (2007) Semi-infinite programming. Eur J Oper Res 180:491–518MathSciNetCrossRefzbMATHGoogle Scholar
  17. López MA, Vercher E (1983) Optimality conditions for nondifferentiable convex semi-infinite Programming. Math Program 27:307–319MathSciNetCrossRefzbMATHGoogle Scholar
  18. Mishra LN (2017) On existence and behavior of solutions to some nonlinear integral equations with applications. Ph.D. Thesis, National Institute of Technology, SilcharGoogle Scholar
  19. Mishra VN (2007) Some problems on approximations of functions in banach spaces. Ph.D. Thesis, Indian Institute of Technology, RoorkeeGoogle Scholar
  20. Mishra SK, Jaiswal M, LeThi HA (2012) Nonsmooth semi-infinite programming problem using limiting subdifferentials. J Global Optim 53:285–296MathSciNetCrossRefzbMATHGoogle Scholar
  21. Mordukhovich BS (2006) Variational analysis and generalized differentiation. I. Basic theory. Springer, BerlinGoogle Scholar
  22. Rockafellar RK, Wets JB (1998) Variational analysis. Springer, BerlinCrossRefzbMATHGoogle Scholar
  23. Vandana (2017) A study of dynamic inventory involving economic ordering of commodity. Ph.D. thesis, Pt. Ravishankar Shukla University Raipur, ChhattisgarhGoogle Scholar
  24. Vandana, Dubey R, Deepmala, Mishra LN, Mishra VN (2018) Duality relations for a class of a multiobjective fractional programming problem involving support functions. Am J Oper Res (in press)Google Scholar

Copyright information

© Shiraz University 2018

Authors and Affiliations

  1. 1.Department of MathematicsIslamic Azad UniversityYazdIran

Personalised recommendations