Encyclopedia of Optimization

2009 Edition
| Editors: Christodoulos A. Floudas, Panos M. Pardalos

Topology of Global Optimization

  • Hubertus Th. Jongen
  • Alina Ruiz Jhones
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-74759-0_684
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Article Outline

Keywords

Introduction, Critical Points, Nondegeneracy

Relations Between KKT Points: Morse Relations

Projected Gradients

Global Gradient Flows: Equality Con-straints Only

Global Gradient Flows: The General Case

See also

References

Keywords

Morse theory Karush–Kuhn–Tucker point Morse relations Euler formula Global optimization Continuous selection of functions Min-max graph Min-max digraph Projected positive gradient Projected negative gradient Ascent flow Descent flow 
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References

  1. 1.
    Agrachev AA, Pallaschke D, Scholtes S (1997) On Morse theory for piecewise smooth functions. J Dynamical and Control Systems 3:449–469MATHMathSciNetGoogle Scholar
  2. 2.
    Brink-Spalink J, Jongen HTh (1979) Morse theory for optimization problems with functions of maximum type. Methods Oper Res 31:121–134MATHMathSciNetGoogle Scholar
  3. 3.
    Craven BD, Gershkovich V, Ralph D (1994) Morse theory and relations between smooth and nonsmooth optimization. In: Glover BM, Jeyakumar V, Anderson GT (eds) Proc. Optimization Miniconference, Univ. Ballarat, pp 39–46Google Scholar
  4. 4.
    Degiovanni M (1990) Homotopical properties of a class of nonsmooth functions. Ann Mat Pura Appl (IV) CLVI:37–71MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dupuis P, Nagurney A (1993) Dynamical systems and variational inequalities. Ann Oper Res 44:9–42MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Filippov AF (1988) Differential equations with discontinuous righthand sides. Kluwer, DordrechtGoogle Scholar
  7. 7.
    Guddat J, Jongen HTh, Rückmann J-J (1986) On stability and stationary points in nonlinear optimization. J Austral Math Soc (Ser B) 28:36–56MATHMathSciNetGoogle Scholar
  8. 8.
    Hirabayashi R, Jongen HTh, Shida M (1994) Euler's formula via potential functions. J Oper Res Japan 37(3):228–231MATHMathSciNetGoogle Scholar
  9. 9.
    Ioffe A, Schwartzman E (1996) Metric critical point theory 1. Morse regularity and homotopic stability of a minimum. J Math Pures Appl 75:125–153MATHMathSciNetGoogle Scholar
  10. 10.
    JongenHTh (1977) Zur Geometrie endlichdimensionaler nichtkonvexer Optimierungsaufgaben. Internat Ser Numer Math 36:111–136MathSciNetGoogle Scholar
  11. 11.
    Jongen HTh, Jonker P, Twilt F (1983) Nonlinear optimization in R n, I. Morse theory, Chebyshev approximation. P. Lang, Frankfurt am MainMATHGoogle Scholar
  12. 12.
    JongenHTh, Jonker P, Twilt F (1986) Nonlinear optimization in R n, II. transversality, flows, parametric aspects. P. Lang, Frankfurt am MainGoogle Scholar
  13. 13.
    JongenHTh, Pallaschke D (1988) On linearization and continuous selections of functions. Optim 19:343–353MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Jongen HTh, Ruiz Jhones A (2000) Nonlinear optimization: On the min-max digraph and global smoothing. In: Ioffe A, Reich S, Shafrir I (eds) Calculus of Variations and Differential Equations, vol 410. Res Notes in Math. Chapman and Hall/CRC, London/Boca Raton, FL, pp 119–135Google Scholar
  15. 15.
    Jongen HTh, Weber G-W (1992) Nonconvex optimization and its structural frontiers. In: Krabs W, Zowe J (eds) Modern Methods of Optimization, vol 378. Lecture Notes Economics and Math Systems, pp 151–203Google Scholar
  16. 16.
    Milnor J (1963) Morse theory. Ann of Math Stud, vol 51. Princeton Univ. Press, PrincetonMATHGoogle Scholar
  17. 17.
    Smale S (1961) On gradient dynamical systems. A-MATH 74(1):199–206MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zank H (1994) Personal communicationGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Hubertus Th. Jongen
    • 1
  • Alina Ruiz Jhones
    • 2
  1. 1.Department Math.Aachen University Technol.AachenGermany
  2. 2.Fac. Math. and Computer Sci.University Havana San Lázaro y LCiudad HabanaCuba