Skip to main content
SpringerLink
Log in
Book cover

Encyclopedia of Optimization pp 3617–3622Cite as

Smoothing Methods for Semi-Infinite Optimization

  • Reference work entry

Article Outline

Introduction

  Smoothing Approaches Motivated by Nonlinear Programming

  Smoothing Approaches for Semi-Infinite Programs

Definitions

  The Extended Mangasarian–Fromovitz Constraint Qualification

  The Reduction Ansatz and Nondegenerate KKT Points

  Mollifiers

Formulation

Conclusions

See also

References

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   2,500.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   2,499.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

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

    Google Scholar 

  2. Evans LC (1998) Partial Differential Equations. American Mathematical Society, Providence, Rhode Island

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  4. Günzel H, Jongen HT (2005) On absorbing cycles in min-max digraphs. J Glob Optim 31:85–92

    Article  MATH  Google Scholar 

  5. Guerra Vasquez F, Günzel H, Jongen HT (2001) On logarithmic smoothing of the maximum function. Ann Oper Res 101:209–220

    Article  MathSciNet  Google Scholar 

  6. Guerra Vasquez F, Rückmann J-J (2002) An approximation of feasible sets in semi-infinite optimization. Top 10:325–336

    Google Scholar 

  7. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  10. Jarre F (1999) Comparing two interior-point approaches for semi-infinite programs, Preprint, University of Trier, http://www.opt.uni-duesseldorf.de/~jarre/papers/semi2.ps

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Jongen HT, Ruiz Jhones A (1999) Nonlinear Optimization: On the min-max digraph and global smoothing. In: Ioffe A, Reich S, Shafrir I (eds) Calculus of Variations and Differential Equations, Chapman and Hall/CRC Research Notes in Mathematics Series, vol 410. CRC Press, Boca Raton, pp 119–135

    Google Scholar 

  13. Jongen HT, Stein O (2004) Constrained global optimization: adaptive gradient flows. In: Floudas CA, Pardalos PM (eds) Frontiers in Global Optimization, Kluwer, Boston, pp 223–236

    Google Scholar 

  14. Jongen HT, Stein O (2006) Smoothing by mollifiers. J Glob Optim. doi:10.1007/s10898-007-9232-3

  15. Jongen HT, Stein O (2006) Smoothing by mollifiers. J Glob Optim. doi:10.1007/s10898-007-9231-4

  16. Jongen HT, Twilt F, Weber G-W (1992) Semi-infinite optimization: structure and stability of the feasible set. J Optim Theory Appl 72:529–552

    Article  MathSciNet  MATH  Google Scholar 

  17. 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 

  18. Lim AEB, Moore JB (1998) A path following algorithm for infinite quadratic programming on a Hilbert space. Discret Continuous Dyn Syst 4:653–670

    Article  MathSciNet  MATH  Google Scholar 

  19. Polak E (1987) On the mathematical foundation of nondifferentiable optimization in engineering design. SIAM Rev 29:21–89

    Article  MathSciNet  Google Scholar 

  20. Stein O (1997) On parametric semi-infinite optimization. Thesis, Shaker, Aachen

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Wetterling W (1970) Definitheitsbedingungen für relative Extrema bei Optimierungs- und Approximationsaufgaben. Num Math 15:122–136

    Article  MathSciNet  MATH  Google Scholar 

  23. Zank H (1994) Personal communication

    Google Scholar 

  24. Zwier G (1987) Structural Analysis in Semi-Infinite Programming. Thesis, University of Twente

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, RWTH Aachen University, Aachen, Germany

    Hubertus T. Jongen

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

    Oliver Stein

Authors
  1. Hubertus T. Jongen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oliver Stein
    View author publications

    You can also search for this author in PubMed Google Scholar

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

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag

About this entry

Cite this entry

Jongen, H., Stein, O. (2008). Smoothing Methods for Semi-Infinite Optimization . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_623

Download citation

Publish with us

Policies and ethics

Search

Navigation