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Generalized Semi-infinite Programming: Optimality Conditions

  • Reference work entry
Encyclopedia of Optimization

  • 282 Accesses

Article Outline

Introduction

Definitions

  Topological Properties

  The Reduction Ansatz

  First-Order Properties of the Feasible Set

  Constraint Qualifications

  Formulation

  Second-Order Optimality Conditions

Conclusions

See also

References

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References

  1. Abadie JM (1967) On the Kuhn–Tucker theorem. In: Abadie J (ed) Nonlinear Programming. Wiley, New York, pp 21–36

    Google Scholar 

  2. Bank B, Guddat J, Klatte D, Kummer B, Tammer K (1983) Non-linear Parametric Optimization. Birkhäuser, Basel

    MATH  Google Scholar 

  3. Bazaraa MS, Sherali HD, Shetty CM (1993) Nonlinear Programming. Theory and Algorithms. Wiley, New York

    MATH  Google Scholar 

  4. Bonnans JF, Shapiro A (2000) Perturbation Analysis of Optimization Problems. Springer, New York

    MATH  Google Scholar 

  5. Cheney EW (1966) Introduction to Approximation Theory. McGraw-Hill, New York

    MATH  Google Scholar 

  6. Danskin JM (1967) The Theory of Max-Min and Its Applications to Weapons Allocation Problems. Springer, New York

    Google Scholar 

  7. Demyanov VF, Vasilev LV (1985) Nondifferentiable Optimization, Optimization Software Inc. Publications Division, New York

    Google Scholar 

  8. Fiacco AV, McCormick GP (1968) Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York

    MATH  Google Scholar 

  9. Gauvin J, Dubeau F (1982) Differential properties of the marginal function in mathematical programming. Math Programm Study 19:101–119

    MathSciNet  Google Scholar 

  10. Goberna MA, López MA (1998) Linear Semi-infinite Optimization. Wiley, Chichester

    MATH  Google Scholar 

  11. Gol'stein EG (1972) Theory of Convex Programming, Translations of Mathematical Monographs, vol 36. American Mathematical Society, Providence, RI

    Google Scholar 

  12. Guerra Vázquez F, Rückmann J-J (2005) Extensions of the Kuhn–Tucker constraint qualification to generalized semi-infinite programming. SIAM J Optim 15(3):926–937

    Article  MathSciNet  MATH  Google Scholar 

  13. Guerra Vázquez F, Rückmann J-J, Stein O (2007) Still G Generalized semi-infinite programming: a tutorial. J Comput Appl Math, DOI: 10.1016/j.cam.2007.02.012

    Google Scholar 

  14. Hettich R, Jongen HT (1978) Semi-infinite programming: conditions of optimality and applications. In: Stoer J (ed) Optimization Techniques, Part 2, Lecture Notes in Control and Information Sciences, vol 7. Springer, Berlin, pp 1–11

    Google Scholar 

  15. Hettich R, Kortanek KO (1993) Semi-infinite programming: theory, methods, and applications. SIAM Rev 35:380–429

    Article  MathSciNet  MATH  Google Scholar 

  16. Hettich R, Still G (1995) Second order optimality conditions for generalized semi-infinite programming problems. Optim 34:195–211

    Article  MathSciNet  MATH  Google Scholar 

  17. Hettich R, Zencke P (1982) Numerische Methoden der Approximation und semi‐infiniten Optimierung. Teubner, Stuttgart

    MATH  Google Scholar 

  18. Hogan WW (1973) Directional derivatives for extremal value functions with applications to the completely convex case. Oper Res 21:188–209

    Article  MathSciNet  MATH  Google Scholar 

  19. Jongen HT, Jonker P, Twilt F (1986) Critical sets in parametric optimization. Math Programm 34:333–353

    Article  MathSciNet  MATH  Google Scholar 

  20. Jongen HT, Rückmann J-J, Stein O (1998) Generalized semi-infinite optimization: a first order optimality condition and examples. Math Programm 83:145–158

    Google Scholar 

  21. Klatte D (1994) Stable local minimizers in semi-infinite optimization: regularity and second-order conditions. J Comp Appl Math 56:137–157

    Article  MathSciNet  MATH  Google Scholar 

  22. Laurent P-J (1972) Approximation et Optimisation. Hermann, Paris

    MATH  Google Scholar 

  23. Lempio F, Maurer H (1980) Differential stability in infinite‐dimensional nonlinear programming. Appl Math Optim 6:139–152

    Article  MathSciNet  MATH  Google Scholar 

  24. Mangasarian OL, Fromovitz S (1967) The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. J Math Anal Appl 17:37–47

    Article  MathSciNet  MATH  Google Scholar 

  25. Polak E (1997) Optimization. Algorithms and Consistent Approximations. Springer, Berlin

    MATH  Google Scholar 

  26. Rockafellar RT (1984) Directional differentiability of the optimal value function in a nonlinear programming problem. Math Program Stud 21:213–226

    MathSciNet  MATH  Google Scholar 

  27. Rückmann J-J, Shapiro A (1999) First-order optimality conditions in generalized semi-infinite programming. J Optim Theory Appl 101:677–691

    Article  MathSciNet  MATH  Google Scholar 

  28. Rückmann J-J, Shapiro A (2001) Second-order optimality conditions in generalized semi-infinite programming. Set-Valued Anal 9:169–186

    Article  MathSciNet  MATH  Google Scholar 

  29. Rückmann J-J, Stein O (2001) On convex lower level problems in generalized semi-infinite optimization. In: Goberna MA, López MA (eds) Semi-infinite Programming – Recent Advances. Kluwer, Dordrecht, pp 121–134

    Google Scholar 

  30. Shapiro A (1984) On optimality conditions in quasidifferentiable optimization. SIAM J Control Optim 22:610–617

    Article  MathSciNet  MATH  Google Scholar 

  31. Stein O (2001) First order optimality conditions for degenerate index sets in generalized semi-infinite programming. Math Oper Res 26:565–582

    Article  MathSciNet  MATH  Google Scholar 

  32. Stein O (2003) Bi-level Strategies in Semi-infinite Programming. Kluwer, Boston

    MATH  Google Scholar 

  33. Stein O (2004) On constraint qualifications in non-smooth optimization. J Optim Theory Appl 121:647–671

    Article  MathSciNet  MATH  Google Scholar 

  34. Stein O, Still G (2000) On optimality conditions for generalized semi-infinite programming problems. J Optim Theory Appl 104:443–458

    Article  MathSciNet  MATH  Google Scholar 

  35. Wetterling W (1970) Definitheitsbedingungen für relative Extrema bei Optimierungs- und Approximationsaufgaben. Numerische Mathematik 15:122–136

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Economics and Business Engineering, University of Karlsruhe, Karlsruhe, Germany

    Oliver Stein

Authors
  1. Oliver Stein
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Editor information

Editors and Affiliations

  1. Department of Chemical Engineering, Princeton University, Princeton, NJ, 08544-5263, USA

    Christodoulos A. Floudas

  2. Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 32611-6595, USA

    Panos M. Pardalos

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© 2008 Springer-Verlag

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Stein, O. (2008). Generalized Semi-infinite Programming: Optimality Conditions . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_213

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