Advertisement

Applied Mathematics and Optimization

, Volume 21, Issue 1, pp 69–88 | Cite as

Global convergence of a semi-infinite optimization method

  • Bradley M. Bell
Article

Abstract

A new algorithm for minimizing locally Lipschitz functions using approximate function values is presented. It yields a method for minimizing semi-infinite exact penalty functions that parallels the trust-region methods used in composite nondifferentiable optimization. A finite method for approximating a semi-infinite exact penalty function is developed. A uniform implicit function theorem is established during this development. An implementation and test results for the approximate penalty function are included.

Keywords

System Theory Mathematical Method Penalty Function Lipschitz Function Global Convergence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bell B (1984) Nonsmooth optimization by successive quadratic programming. Ph.D. Dissertation, University of Washington, SeattleGoogle Scholar
  2. 2.
    Burke JV (1985) Descent methods for composite nondifferentiable optimization problems. Math Programming 33:260–279Google Scholar
  3. 3.
    Clarke FH (1976) A new approach to Lagrange multipliers. Math Oper Res 1:165–174Google Scholar
  4. 4.
    Conn AR, Gould NIM (1987) An exact penalty function for semi-infinite programming. Math Programming 37:19–40Google Scholar
  5. 5.
    Coope ID, Watson GA (1985) A projected Lagrangian algorithm for semi-infinite programming. Math Programming 32:337–356Google Scholar
  6. 6.
    Fletcher R (1981) Practical Methods of Optimization. Wiley, New YorkGoogle Scholar
  7. 7.
    Furukawa N (1983) Optimality conditions in nondifferentiable programming and applications to best approximations. Appl Math Optim 9:337–371Google Scholar
  8. 8.
    Gonzaga C, Polak E, Trahan R (1980) An improved algorithm for optimization problems with functional inequality constraints. IEEE Trans Automat Control 25:49–54Google Scholar
  9. 9.
    Madsen K (1986) Minimization of non-linear approximation functions. Ph.D. Dissertation, Institute for Numerical Analysis, Technical University of DenmarkGoogle Scholar
  10. 10.
    Marquardt DW (1963) An algorithm for least squares estimation of nonlinear parameters. SIAM J Appl Math 11:431–440Google Scholar
  11. 11.
    Mayne DQ, Polak E, Trahan R (1979) An outer approximation algorithm for computer-aided design problems. J. Optim Theory Appl 28:331–352Google Scholar
  12. 12.
    Morea JJ (1978) The Levenberg-Marquardt algorithm: implementation and theory. In: Watson, GA (ed) Lecture Notes in Mathematics, vol 630. Springer-Verlag, New York, pp 105–116Google Scholar
  13. 13.
    Polak E, Mayne DQ (1976) An algorithm for optimization problems with functional inequality constraints. IEEE Trans Automat Control 21:184–193Google Scholar
  14. 14.
    Polak E, Tits AL (1982) A recursive quadratic programming algorithm for semi-infinite optimization problems. Appl. Math Optim 8:325–349Google Scholar
  15. 15.
    Powell MJD (1986) Convergence properties of algorithms for nonlinear optimization. SIAM Rev 28:487–500Google Scholar
  16. 16.
    Rockafellar RT (1970) Convex Analysis. Princeton University Press, Princeton, NJGoogle Scholar
  17. 17.
    Rosenberg E (1984) Exact penalty functions and stability in locally Lipschitz programming. Math Programming 30:340–355Google Scholar
  18. 18.
    Sorensen DC (1982) Newton's method with a model trust region modification. SIAM J Numer Anal 19:409–426Google Scholar
  19. 19.
    Tapia RA (1971) The differentiation and integration of nonlinear operators. In: Rall LB (ed) Nonlinear Functional Analysis and Applications. Academic Press, New York, pp 45–101Google Scholar
  20. 20.
    Wets RJB (1980) Convergence of convex functions, variational inequalities and convex optimization problems. In: Cottle RW, Giannessi F, Lions JL (eds) Variational Inequalities and Complementarity Problems. Wiley, New York, pp 375–403Google Scholar
  21. 21.
    Womersley RS (1985) Local properties of algorithms for minimizing nonsmooth composite functions. Math Programming 32:69–89Google Scholar
  22. 22.
    Yuan Y (1985) Conditions for convergence of trust region algorithms for nonsmooth optimization. Math Programming 31:220–228Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Bradley M. Bell
    • 1
  1. 1.Applied Physics Laboratory, College of Ocean and Fishery SciencesUniversity of WashingtonSeattleUSA

Personalised recommendations