Skip to main content
Log in

Global convergence in infeasible-interior-point algorithms

  • Published:
Mathematical Programming Submit manuscript

Abstract

This paper presents a wide class of globally convergent interior-point algorithms for the nonlinear complementarity problem with a continuously differentiable monotone mapping in terms of a unified global convergence theory given by Polak in 1971 for general nonlinear programs. The class of algorithms is characterized as: Move in a Newton direction for approximating a point on the path of centers of the complementarity problem at each iteration. Starting from a strictly positive but infeasible initial point, each algorithm in the class either generates an approximate solution with a given accuracy or provides us with information that the complementarity problem has no solution in a given bounded set. We present three typical examples of our interior-point algorithms, a horn neighborhood model, a constrained potential reduction model with the use of the standard potential function, and a pure potential reduction model with the use of a new potential function.

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

Access this article

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. R.W. Cottle, “Note on a fundamental theorem on quadratic programming,”SIAM Journal on Applied Mathematics 12 (1964) 663–665.

    Google Scholar 

  2. A.V. Fiacco and G.P. McCormick,Nonlinear Programming: Sequential Unconstrained Minimization Techniques (John Wiley & Sons, New York, 1968).

    Google Scholar 

  3. K.R. Frish, “The logarithmic potential method of convex programming,” Technical report, University Institute of Economics, Oslo, Norway, 1955.

    Google Scholar 

  4. O. Güler, “Existence of interior points and interior paths in nonlinear monotone complementarity problems,”Mathematics of Operations Research 18 (1993) 128–147.

    Google Scholar 

  5. N. Karmarkar, “A new polynomial-time algorithm for linear programming,”Combinatorica 4 (1984) 373–395.

    Google Scholar 

  6. M. Kojima, N. Megiddo and S. Mizuno, “A general framework of continuation methods for complementarity problems,”Mathematics of Operations Research 18 (1993) 945–963.

    Google Scholar 

  7. M. Kojima, N. Megiddo and S. Mizuno, “A primal-dual infeasible-interior-point algorithm for linear programming,”Mathematical Programming 61 (1993) 263–280.

    Google Scholar 

  8. M. Kojima, N. Megiddo and T. Noma, “Homotopy continuation methods for nonlinear complementarity problems,”Mathematics of Operations Research 16 (1991) 754–774.

    Google Scholar 

  9. M. Kojima, N. Megiddo, T. Noma and A. Yoshise,A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Lecture Notes in Computer Science 538 (Springer-Verlag, New York, 1991).

    Google Scholar 

  10. M. Kojima, N. Megiddo and Y. Ye, “An interior point potential reduction algorithm for the linear complementarity problem,”Mathematical Programming 54 (1992) 267–279.

    Google Scholar 

  11. M. Kojima, S. Mizuno and A. Yoshise, “A primal-dual interior point algorithm for linear programming,” in: N. Megiddo, ed.,Progress in Mathematical Programming, Interior-Point and Related Methods (Springer-Verlag, New York, 1989) 29–47.

    Google Scholar 

  12. M. Kojima, S. Mizuno and A. Yoshise, “A polynomial-time algorithm for a class of linear complementary problems,”Mathematical Programming 44 (1989) 1–26.

    Google Scholar 

  13. M. Kojima, S. Mizuno and A. Yoshise, “An O\((\sqrt {nL} )\) iteration potential reduction algorithm for linear complementarity problems,”Mathematical Programming 50 (1991) 331–342.

    Google Scholar 

  14. M. Kojima, S. Mizuno and A. Yoshise, “A little theorem of the big in interior point algorithms,”Mathematical Programming, 59 (1993) 361–375.

    Google Scholar 

  15. K.O. Kortanek and J. Zhu, “A polynomial algorithm for convex programming problems satisfying a scaled Lipschitz condition,” Working Paper Series No. 90-17, Dept. of Management Sciences, College of Business Administration, The University of Iowa, Iowa City, Iowa 52242, 1990.

    Google Scholar 

  16. I.J. Lustig, “Feasibility issues in a primal-dual interior-point method for linear programming,”Mathematical Programming 49 (1990/91) 145–162.

    Google Scholar 

  17. L. McLinden, “The complementarity problem for maximal monotone multifunctions,” in: R.W. Cottle, F. Giannessi and J.L. Lions, eds.,Variational Inequalities and Complementarity Problems (John Wiley & Sons, New York, 1980) 251–270.

    Google Scholar 

  18. K. A. McShane, C.L. Monma and D.F. Shanno, “An implementation of a primal-dual interior point method for linear programming,”ORSA Journal on Computing 1 (1989) 70–83.

    Google Scholar 

  19. N. Megiddo, “A monotone complementarity problem with feasible solutions but no complementary solutions,”Mathematical Programming 12 (1977) 131–132.

    Google Scholar 

  20. N. Megiddo, “Pathways to the optimal set in linear programming,” in: N. Megiddo, ed.,Progress in Mathematical Programming, Interior-Point and Related Methods, (Springer-Verlag, New York, 1989) 131–158.

    Google Scholar 

  21. S. Mizuno, “Polynomiality of Kojima—Megiddo—Mizuno infeasible interior point algorithm for linear programming,” Technical report, The Institute of Statistical Mathematics, Minami-Azabu, Minato-ku, Tokyo, 106, Japan, 1992.

    Google Scholar 

  22. S. Mizuno, M. Kojima and M.J. Todd, “Infeasible-interior-point primal-dual potential-reduction algorithms for linear programming,”Siam Journal on Optimization, to appear.

  23. S. Mizuno and A. Nagasawa, “A primal-dual affine scaling potential reduction algorithm for linear programming,”Mathematical Programming 62 (1993) 119–131.

    Google Scholar 

  24. S. Mizuno, M.J. Todd and Y. Ye, “On adaptive-step primal-dual interior-point algorithms for linear programming,”Mathematics of Operations Research 18 (1993) 964–981.

    Google Scholar 

  25. S. Mizuno, M.J. Todd and Y. Ye, “A surface of analytic centers and infeasible-interior-point algorithms for linear programming,” Technical Report, School of Operations Research and Industrial Engineering, Cornell University (Ithaca, NY 14853, December 1992).

    Google Scholar 

  26. R.D.C. Monteiro, “A globally convergent primal-dual interior point algorithm for convex programming,” Technical report, Systems and Industrial Engineering, University Arizona, Tucson, AZ 85721, 1991.

    Google Scholar 

  27. R.D.C. Monteiro and I. Adler, “Interior path following primal-dual algorithms. Part I: linear programming,”Mathematical Programming 44 (1989) 27–41.

    Google Scholar 

  28. R.D.C. Monteiro and I. Adler, “Interior path following primal-dual algorithms. Part II: convex quadratic programming,”Mathematical Programming 44 (1989) 43–66.

    Google Scholar 

  29. R.D.C. Monteiro and I. Adler, “An extension of Karmarkar type algorithm to a class of convex separable programming problems with global linear rate of convergence,”Mathematics of Operations Research 15 (1990) 408–422.

    Google Scholar 

  30. R.D.C. Monteiro and S.J. Wright, “A globally and superlinearly convergent potential reduction interior point method for convex programming,” Technical report, Systems and Industrial Engineering, University Arizona, Tucson, AZ 85721, 1992.

    Google Scholar 

  31. J.J. More, Class of functions and feasibility conditions in nonlinear complementarity problem,Mathematical Programming 6 (1974) 327–338.

    Google Scholar 

  32. T. Noma, “A globally convergent iterative algorithm for complementarity problems — a modification of interior point algorithms for linear complementarity problems —”, Dr. Thesis, Dept. of Systems Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152, Japan, 1991.

    Google Scholar 

  33. E. Polak,Computational Methods in Optimization: Approach (Academic Press, New York, 1971).

    Google Scholar 

  34. F.A. Potra, “An infeasible interior-point predictor-corrector algorithm for linear programming,” Technical report, Dept. of Mathematics, The University of Iowa, Iowa City, Iowa 52242, 1992.

    Google Scholar 

  35. F.A. Potra and Y. Ye, “Interior point methods for nonlinear complementarity problems,” Technical report, Dept. of Mathematics, The University of Iowa, Iowa City, Iowa 52242, 1991.

    Google Scholar 

  36. J. Renegar, “Incorporating condition measures into the complexity theory of linear programming,” Technical report, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853-3801, 1993.

    Google Scholar 

  37. G. Sonnevend, “An ‘analytical centre’ for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming,” in:Lecture Notes in Control and Information Sciences 84 (Springer, New York, 1985) 866–876.

    Google Scholar 

  38. K. Tanabe, “Complementarity-enforcing centered Newton method for mathematical programming,” in: K. Tone, ed.,New Methods for Linear Programming (The Institute of Statistical Mathematics, Minamiazabu, Minato-ku, Tokyo 106, Japan, 1987) 118–144.

    Google Scholar 

  39. K. Tanabe, “Centered Newton method for mathematical programming,” in: M. Iri and K. Yajima, eds,System Modelling and Optimization (Springer-Verlag, New York, 1988) 197–206.

    Google Scholar 

  40. M.J. Todd, “Projected scaled steepest descent in Kojima—Mizuno—Yoshise's potential reduction algorithm for the linear complementarity problem,” Technical Report No. 950, School of Operations Research and Industrial Engineering, College of Engineering, Cornell University, Ithaca, New York 14853-3801, 1990.

    Google Scholar 

  41. M.J. Todd and Y. Ye, “A centered projective algorithm for linear programming,”Mathematics of Operations Research 15 (1990) 508–529.

    Google Scholar 

  42. L. Tunçel, “Constant potential primal-dual algorithm: a framework,” Working Paper, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853, July 1992.

    Google Scholar 

  43. Y. Ye and K. Anstreicher, “On quadratic and O\((\sqrt {nL} )\) convergence of a predictor- corrector algorithm for LCP,” Research Report, Dept. of Management Sciences, University of Iowa, Iowa City, IA 52242, November 1991.

    Google Scholar 

  44. Y. Ye, O. Güler, R.A. Tapia and Y. Zhang, “A quadratically convergent O\((\sqrt {nL} )\)-iteration algorithm for linear programming,” TR91-26, Dept. of Mathematical Sciences, Rice University, Houston, TX 77251-1892, August 1991.

    Google Scholar 

  45. Y. Ye, M.J. Todd and S. Mizuno, “An O\((\sqrt {nL} )\)-iteration homogeneous and self-dual linear programming algorithm,” Research Report, Dept. of Management Sciences, University of Iowa, Iowa City, IA 52242, June 1992.

    Google Scholar 

  46. W.I. Zangwill,Nonlinear Programming: A Unified Approach (Prentice-Hall, Englewood Cliffs, N. J., 1969).

    Google Scholar 

  47. Y. Zhang, “On the convergence of an infeasible interior-point algorithm for linear programming and other problems,”Siam Journal on Optimization, to appear.

  48. J. Zhu, “A path following algorithm for a class of convex programming problems,”Zeitschrift für Operations Research — Methods and Models of Operations Research 36 (1992) 539–577.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Research supported in part by Grant-in-Aids for Co-Operative Research (03832017) of the Japan Ministry of Education, Science and Culture.

Corresponding author.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kojima, M., Noma, T. & Yoshise, A. Global convergence in infeasible-interior-point algorithms. Mathematical Programming 65, 43–72 (1994). https://doi.org/10.1007/BF01581689

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01581689

Keywords

Navigation