Skip to main content
SpringerLink
Log in

On Superlinear Convergence of Infeasible Interior-Point Algorithms for Linearly Constrained Convex Programs

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

This note derives bounds on the length of the primal-dual affinescaling directions associated with a linearly constrained convexprogram satisfying the following conditions: 1) the problem has asolution satisfying strict complementarity, 2) the Hessian of theobjective function satisfies a certain invariance property. Weillustrate the usefulness of these bounds by establishing thesuperlinear convergence of the algorithm presented in Wright andRalph [22] for solving the optimality conditions associatedwith a linearly constrained convex program satisfying the aboveconditions.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. C. Gonzaga and R. Tapia, On the convergence of the Mizuno-Todd-Ye algorithm to the analytic center of the solution set, SIAM J. Optim., 7, 1997, pp. 47–65.

    Google Scholar 

  2. M. Kojima, N. Megiddo and S. Mizuno, A primal-dual infeasible-interior-point algorithm for linear programming, Mathematical Programming, 61, 1993, pp. 263–280.

    Google Scholar 

  3. M. Kojima, N. Megiddo and T. Noma, Homotopy continuation methods for nonlinear complementarity problems, Mathematics of Operations Research, 16, 1991, pp. 754–774.

    Google Scholar 

  4. S. Mehrotra, Quadratic convergence in a primal-dual method, Mathematics of Operations Research, 18, 1993, pp. 741–751.

    Google Scholar 

  5. S. Mizuno, Polynomiality of infeasible-interior-point algorithms for linear programming, Mathematical Programming, 67, 1994, pp. 109–119.

    Google Scholar 

  6. R. D. C. Monteiro, T. Tsuchiya and Y. Wang, A Simplified Global Convergence Proof of the Affine Scaling Algorithm, Annals of Operations Research, 47, 1993, 443–482.

    Google Scholar 

  7. R. D. C. Monteiro and Y. Wang, Trust region affine scaling algorithms for linearly constrained convex and concave programs, Technical Report, School of ISyE, Georgia Institute of Technology, Atlanta, GA 30332, USA, June, 1995. To appear in Mathematical Programming.

    Google Scholar 

  8. R. D. C. Monteiro and S. Wright, " A globally and superlinearly convergent potential reduction interior point method for convex programming, Technical Report, 92-13, Dept. of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA, July, 1992.

    Google Scholar 

  9. R. D. C. Monteiro and S. Wright, Local convergence of interior-point algorithms for degenerate monotone LCPs, Computational Optimization and Applications,3, 1994, pp. 131–155.

    Google Scholar 

  10. R. D. C. Monteiro and S. Wright, Superlinear primal-dual affine scaling algorithms for LCP, Mathematical Programming, 69, 1995, pp. 311–333.

    Google Scholar 

  11. R. D. C. Monteiro and S. Wright, A superlinear infeasible-interior-point affine scaling algorithm for LCP, SIAM Journal on Optimization, 6, 1996, pp. 1–18.

    Google Scholar 

  12. F. A. Potra, An O(nL) infeasible-interior-point algorithm for LCP with quadratic convergence, Ann. of Oper. Res., 62, 1996, pp. 81–102.

    Google Scholar 

  13. F. A. Potra, A quadratically convergent predictor-corrector method for solving linear programs from infeasible starting points, Mathematical Programming, 67, 1994, pp. 383–406.

    Google Scholar 

  14. F. A. Potra and Y. Ye, Interior point methods for nonlinear complementarity problems, J. Optim. Theory Appl., 88, 1996, pp. 617–642.

    Google Scholar 

  15. F. A. Potra and Y. Ye, A quadratically convergent polynomial algorithm for solving entropy optimization problems, SIAM Journal on Optimization, 3, 1993, pp. 843–861.

    Google Scholar 

  16. E. M. Simantiraki and D. F. Shanno, An infeasible-interior-point method for linear complementarity problems, RUTCOR Research Report, RRR 7–95, Rutgers Center for Operations Research, Rutgers University, New Brunswick, NJ 08903, USA, 1995.

    Google Scholar 

  17. P. Tseng and Z. Q. Luo, On the convergence of the affine-scaling algorithm, Mathematical Programming, 56, 1992, pp. 301–319.

    Google Scholar 

  18. Tunçel, L., Constant potential primal-dual algorithms: A framework, Mathematical Programming, 66, 1994, pp. 145–159.

    Google Scholar 

  19. S. Wright, A path-following infeasible-interior-point algorithm for linear complementarity problems, Optimization Methods and Software, 2, 1993, pp. 79–106.

    Google Scholar 

  20. S. Wright,A path-following interior-point algorithm for linear and quadratic optimization problems, Ann. Oper. Res., 62, 1996, pp. 103–130.

    Google Scholar 

  21. S. Wright, An infeasible interior point algorithm for linear complementarity problems, Mathematical Programming, 67, 1994, 29–51.

    Google Scholar 

  22. S. Wright and D. Ralph, A superlinear infeasible-interior-point algorithm for monotone complementarity problems, Math. Oper. Res., 21, 1996, pp. 815–838.

    Google Scholar 

  23. Y. Ye, On the Q-order of convergence of interior-point algorithms for linear programming, in Proceedings of the 1992 Symposium on Applied Mathematics, F. Wu, ed., Institute of Applied Mathematics, Chinese Academy of Sciences, 1992, pp. 534–543.

  24. Y. Ye and K. Anstreicher, On quadratic and O(pnL) convergence of a predictor-corrector algorithm for LCP, Mathematical Programming, 62, 1993, pp. 537–551.

    Google Scholar 

  25. Y. Ye, O. G¨uler, R. A. Tapia and Y. Zhang, A quadratically convergent O(pnL)-iteration algorithm for linear programming, Mathematical Programming, 59, 1993, pp. 151–162.

    Google Scholar 

  26. Y. Zhang, On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem, SIAM Journal on Optimization, 4, 1994, pp. 208–227.

    Google Scholar 

  27. Y. Zhang and R. A. Tapia, Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited, Journal of Optimization Theory and Applications, 73, 1992, pp. 229–242.

    Google Scholar 

  28. Y. Zhang and R. A. Tapia, A superlinearly convergent polynomial primal-dual interior-point algorithm for linear programming, SIAM Journal on Optimization, 3, 1993, pp. 118–133.

    Google Scholar 

  29. Y. Zhang, R. A. Tapia and J. E. Dennis, On the superlinear and quadratic convergence of primal-dual interior point linear programming algorithms, SIAM Journal on Optimization, 2, 1992, pp. 304–324.

    Google Scholar 

  30. Y. Zhang and D. Zhang, On polynomiality of the Mehrotra-type predictor-corrector interior-point algorithms, Mathematical Programming, 68, 1995, pp. 303–318.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 30332

    Renato D.C. Monteiro & Fangjun Zhou

Authors
  1. Renato D.C. Monteiro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fangjun Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Monteiro, R.D., Zhou, F. On Superlinear Convergence of Infeasible Interior-Point Algorithms for Linearly Constrained Convex Programs. Computational Optimization and Applications 8, 245–262 (1997). https://doi.org/10.1023/A:1008623505672

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008623505672

Search

Navigation