Journal of Optimization Theory and Applications

, Volume 91, Issue 2, pp 389–418 | Cite as

Convergence analysis of norm-relaxed method of feasible directions

  • J. Korycki
  • M. Kostreva
Contributed Papers

Abstract

This paper gives a complete treatment of the asymptotic rate of convergence for a class of feasible directions methods, including those studied by Pironneau and Polak and by Cawood and Kostreva. Rate estimates of Pironneau and Polak are sharpened in an analysis which shows the dependence on certain parameters of the direction-finding subproblem and the problem functions. Special cases of interior optimal solution, linear constraints, and fixed matrix norm are analyzed in detail. Numerical verification is provided.

Key Words

Nonlinear programming method of feasible directions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Zoutendijk, G.,Methods of Feasible Directions, Elsevier, Amsterdam, Holland, 1960.Google Scholar
  2. 2.
    Topkis, D. M., andVeinott, A. F.,On the Convergence of Some Feasible Direction Algorithms for Nonlinear Programming, SIAM Journal on Control, Vol. 5, pp. 268–279, 1967.Google Scholar
  3. 3.
    Pironneau, O., andPolak, E.,On the Convergence of Certain Methods of Centers, Mathematical Programming, Vol. 2, pp. 230–257, 1972.Google Scholar
  4. 4.
    Pironneau, O., andPolak, E.,Rate of Convergence of a Class of Methods of Feasible Directions, SIAM Journal on Numerical Analysis, Vol. 10, pp. 161–173, 1973.Google Scholar
  5. 5.
    Vanderplaats, G. N.,Numerical Optimization Techniques for Engineering Design, McGraw Hill, New York, New York, 1984.Google Scholar
  6. 6.
    Vanderplaats, G. N.,Efficient Feasible Directions Algorithm for Design Synthesis, AIAA Journal, Vol. 22, pp. 1633–1640, 1984.Google Scholar
  7. 7.
    Vanderplaats, G. N.,DOT/DOC Users Manual, Vanderplaats, Miura, and Associates, 1993.Google Scholar
  8. 8.
    Meyer, G. L.,An Efficient Method of Feasible Directions, SIAM Journal on Control and Optimization, Vol. 21, pp. 153–162, 1983.Google Scholar
  9. 9.
    Polak, E., andTishyadhigama, S.,New Convergence Theorems for a Class of Feasible Directions Algorithms, Journal of Optimization Theory and Applications, Vol. 37, pp. 33–44, 1982.Google Scholar
  10. 10.
    Tits, A. L., Nye, W. T., andSangiovanni-Vincentelli, A. L.,Enhanced Methods of Feasible Directions for Engineering Design Problems, Journal of Optimization Theory and Applications, Vol. 51, pp. 475–504, 1986.Google Scholar
  11. 11.
    Allwright, J. C.,A Feasible Direction Algorithm for Convex Optimization: Global Convergence Rates, Journal of Optimization Theory and Applications, Vol. 30, pp. 1–18, 1980.Google Scholar
  12. 12.
    Chaney, R. W.,On the Pironneau-Polak Method of Centers, Journal of Optimization Theory and Applications, Vol. 20, pp. 269–295, 1976.Google Scholar
  13. 13.
    Chaney, R. W.,On the Rate of Convergence of Some Feasible Direction Algorithms, Journal of Optimization Theory and Applications, Vol. 20, pp. 297–313, 1976.Google Scholar
  14. 14.
    Cawood, M. E., andKostreva, M. M.,Norm-Relaxed Method of Feasible Directions for Solving Nonlinear Programming Problems, Journal of Optimization Theory and Applications, Vol. 83, 1994.Google Scholar
  15. 15.
    Horn, R. A., andJohnson, C. R.,Matrix Analysis, Cambridge University Press, Cambridge, England, 1985.Google Scholar
  16. 16.
    Polak, E.,Computational Methods in Optimization, Academic Press, New York, New York, 1971.Google Scholar
  17. 17.
    Huard, P.,Resolution of Mathematical Programming with Nonlinear Constraints by the Method of Centers, Nonlinear Programming, Edited by J. Abadie, North Holland, Amsterdam, Holland, pp. 207–219 1967.Google Scholar
  18. 18.
    Kostreva, M. M.,Generalization of Murty's Direct Algorithm to Linear and Convex Quadratic Programming, Journal of Optimization Theory and Applications, Vol. 62, pp. 63–76, 1989.Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • J. Korycki
    • 1
  • M. Kostreva
    • 1
  1. 1.Department of Mathematical SciencesClemson UniversityClemson

Personalised recommendations