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The Kantorovich Theorem and interior point methods

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Abstract.

The Kantorovich Theorem is a fundamental tool in nonlinear analysis which has been extensively used in classical numerical analysis. In this paper we show that it can also be used in analyzing interior point methods. We obtain optimal bounds for Newton’s method when relied upon in a path following algorithm for linear complementarity problems.

Given a point z that approximates a point z(τ) on the central path with complementarity gap τ, a parameter θ ∈ (0,1) is determined for which the point z satisfies the hypothesis of the affine invariant form of the Kantorovich Theorem, with regards to the equation defining z((1−θ)τ). The resulting iterative algorithm produces a point with complementarity gap less than ε in at most Newton steps, or simplified Newton steps, where ε0 is the complementarity gap of the starting point and n is the dimension of the problem. Thus we recover the best complexity results known to date and, in addition, we obtain the best bounds for Newton’s method in this context.

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References

  1. Alefeld, G.E., Potra, F.A., Shen, Z.: On the existence theorems of Kantorovich, Moore and Miranda. Computing Supplementum 15, 21–28 (2001)

    Google Scholar 

  2. Amer, S.M.: On solution of nonlinear singular integral equations with shift in generalized Hölder space. Chaos Solitons Fractals 12 (7), 1323–1334 (2001)

    Article  Google Scholar 

  3. Begehr, H., Efendiev, M.A.: On the asymptotics of meromorphic solutions for nonlinear Riemann-Hilbert problems. Math. Proc. Cambridge Philos. Soc. 127 (1), 159–172 (1999)

    Article  Google Scholar 

  4. Cottle, R.W., Pang, J.-S., Stone, R.E.: m The Linear Complementarity Problem. Academic Press, Boston, MA, 1992

  5. De Pascale, E., Zabrejko, P.P.: New convergence criteria for the Newton-Kantorovich method and some applications to nonlinear integral equations. Rend. Sem. Mat. Univ. Padova 100, 211–230 (1998)

    Google Scholar 

  6. Dennis, J.E. Jr.: On the Kantorovich hypothesis for Newton’s method. SIAM J. Numer. Anal. 6, 493–507 (1969)

    Google Scholar 

  7. Dennis, J.E. Jr.: On the convergence of Newton-like methods. In: Numerical methods for nonlinear algebraic equations (Proc. Conf., Univ. Essex, Colchester, 1969), pp. 163–181. Gordon and Breach, London, 1970

  8. Deuflhard, P., Heindl, G.: Affine invariant convergence theorems for Newton’s method and extensions to related methods. SIAM J. Numer. Anal. 16, 1–10 (1979)

    MathSciNet  MATH  Google Scholar 

  9. Ezquerro, J.A., Hernández, M.A.: Multipoint super-Halley type approximation algorithms in Banach spaces. Numer. Funct. Anal. Optim., 21 (7-8), 845–858 (2000)

    Google Scholar 

  10. Ezquerro, J.A., Hernández, M.A.: A special type of Hammerstein integral equations. Int. Math. J. 1 (6), 557–566 (2002)

    Google Scholar 

  11. Ferreira, O.P., Svaiter, B.F.: Kantorovich’s theorem on Newton’s method in Riemannian manifolds. J. Complexity 18 (1), 304–329 (2002)

    Article  Google Scholar 

  12. Gragg, W.B., Tapia, R.A.: Optimal error bounds for the Newton-Kantorovich theorem. SIAM J. Numer. Anal. 11, 10–13 (1974)

    Google Scholar 

  13. Grigat, E., Sachs, G.: Predictor-corrector continuation method for optimal control problems. In: Variational calculus, optimal control and applications (Trassenheide, 1996), volume 124 of Internat. Ser. Numer. Math., pp. 223–232. Birkhäuser, Basel, 1998

  14. Jansen, B., Roos, C., Terlaky, T., Vial, J.-P.: Primal-dual algorithms for linear programming based on the logarithmic barrier method. J. Optim. Theory Appl. 83 (1), 1–26 (1994)

    Google Scholar 

  15. Jansen, B., Roos, C., Terlaky, T., Vial, J.-Ph.: Primal-dual target-following algorithms for linear programming. Ann. Oper. Res. 62, 197–231 (1996) Interior point methods in mathematical programming

    Google Scholar 

  16. Kantorovich, L.: On Newton’s method for functional equations (Russian). Dokl. Akad. Nauk. SSSR 59, 1237–1240 (1948)

    Google Scholar 

  17. Kantorovich, L.V., Akilov, G.P.: Functional analysis in normed spaces, volume 46 of International Series of Monographs in Pure and Applied Mathematics. Pergamon Press, Oxford, 1964. Translated from the Russian by D. E. Brown and Dr. A. P. Robertson

  18. Kojima, M., Megiddo, N., Noma, T., Yoshise, A.: A unified approach to interior point algorithms for linear complementarity problems, volume 538 of Lecture Notes in Comput. Sci. Springer-Verlag, New York, 1991

  19. Laumen, M.: A Kantorovich theorem for the structured PSB update in Hilbert space. J. Optim. Theory Appl. 105 (2), 391–415 (2000)

    Article  Google Scholar 

  20. Matveev, A.F., Yunganns, P.: On the construction of an approximate solution of a nonlinear integral equation of permeable profile. Differ. Uravn. 33 (9), 1242–1252, 1295 (1997)

    Google Scholar 

  21. Mayer, J.: A generalized theorem of Miranda and the theorem of Newton-Kantorovich. Numer. Funct. Anal. Optim. 23(3-4), 333–357 (2002)

    Google Scholar 

  22. Miel, G.: An updated version of the Kantorovich theorem for Newton’s method. Computing 27, 237 (1981)

    Google Scholar 

  23. Miel, G.J.: The Kantorovich theorem with optimal error bounds. American Mathematical Monthly 86 (3), 212–215 (1979)

    Google Scholar 

  24. Moret, I.: A note on Newton type iterative methods. Computing 33, 65–73 (1984)

    Google Scholar 

  25. Moret, I.: On a general iterative scheme for Newton-type methods. Numerical Functional Analysis and Optimization 9, 1115–1137 (1987)

    Google Scholar 

  26. Moret, I.: A Kantorovich-type theorem for inexact Newton methods. Numer. Funct. Anal. Optim., 10 (3-4), 351–365 (1989)

    Google Scholar 

  27. Mukaidani, H., Shimomura, T., Mizukami, K.: Asymptotic expansions and a new numerical algorithm of the algebraic Riccati equation for multiparameter singularly perturbed systems. J. Math. Anal. Appl. 267 (1), 209–234 (2002)

    Article  Google Scholar 

  28. Nagatou, K., Yamamoto, N., Nakao, M.T.: An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness. Numer. Funct. Anal. Optim. 20 (5-6), 543–565 (1999)

    Google Scholar 

  29. Nashed, M.Z., Chen, X.: Convergence of Newton-like methods for singular operator equations using outer inverses. Numer. Math. 66 (2), 235–257 (1993)

    Google Scholar 

  30. Nayakkankuppam, M.V., Overton, M.L.: Conditioning of semidefinite programs. Math. Program. 85 (3, Ser. A), 525–540 (1999)

  31. Nesterov, Y., Nemirovsky, A.: Interior Point Polynomial Methods in Convex Programming. SIAM Publications. SIAM, Philadelphia, 1994

  32. Ortega, J.M.: The Newton-Kantorovich theorem. Amer. Math. Monthly 75, 658–660 (1968)

    Google Scholar 

  33. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970

  34. Ostrowski, A.M.: Solution of Equations in Euclidean and Banach Spaces. Academic Press, New York, 1973

  35. Potra, F.A.: An error analysis for the secant method. Numer. Math. 38, 427–445 (1982)

    Google Scholar 

  36. Potra, F.A.: On the aposteriori error estimates for Newton’s method. Beitraege Numer. Math. 12, 125–138 (1984)

    Google Scholar 

  37. Potra, F.A.: Sharp error bounds for a class of Newton–like methods. Libertas Mathematica 5, 71–84 (1985)

    Google Scholar 

  38. Potra, F.A.: A path-following method for linear complementarity problems based on the affine invariant Kantorovich theorem. ZIB-Report 00-30, Konrad-Zuse-Zentrum, Berlin, August 2000

  39. Potra, F.A., Pták, V.: Sharp error bounds for Newton’s process. Numer. Math. 34, 63–72 (1980)

    Google Scholar 

  40. Potra, F.A., Pták, V.: Nondiscrete induction and iterative processes. Number 103 in Research Notes in Mathematics. John Wiley & Sons, Boston–London–Melbourne, 1984

  41. Potra, F.A., Wright, S.J.: Interior-point methods. Journal of Computational and Applied Mathematics 124, 281–302 (2000)

    Article  Google Scholar 

  42. Rall, L.B.: Computational solution of nonlinear operator equations. Krieger Huntington, New York, 1979

  43. Renegar, J.: A polynomial-time algorithm, based on Newton’s method, for linear programming. Math. Programming 40 (1,Ser. A), 59–93 (1988)

  44. Renegar, J.: A mathematical view of interior-point methods in convex optimization. MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001

  45. Renegar, J., Shub, M.: Unified complexity analysis for Newton LP methods. Math. Programming 53 (1, Ser. A), 1–16 (1992)

  46. Roos, C., Vial, J.-Ph., Terlaky, T.: Theory and algorithms for linear optimization: an interior point approach. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley and Sons, 1997

  47. Smale, S.: Newton’s method estimates from data at one point. In: The merging of disciplines: new directions in pure, applied, and computational mathematics (Laramie, Wyo., 1985), pp. 185–196. Springer, New York, 1986

  48. Stoffer, D., Palmer, K.J.: Rigorous verification of chaotic behaviour of maps using validated shadowing. Nonlinearity 12 (6), 1683–1698 (1999)

    Article  Google Scholar 

  49. Tapia, R.A.: The Kantorovich theorem for Newton’s method. American Mathematical Monthly 78, 389–392 (1971)

    Google Scholar 

  50. Tsuchiya, T.: An application of the Kantorovich theorem to nonlinear finite element analysis. Numer. Math. 84 (1), 121–141 (1999)

    Article  Google Scholar 

  51. Wang, X., Li, C., Lai, M.-J.: A unified convergence theory for Newton-type methods for zeros of nonlinear operators in Banach spaces. BIT 42 (1), 206–213 (2002)

    Article  Google Scholar 

  52. Wright, S.J.: Primal–Dual Interior–Point Methods. SIAM Publications, Philadephia, 1997

  53. Yamamoto, T.: A unified derivation of several error bounds for Newton’s process. Journal of Computation and Applied Mathematics 12, 179–191 (1985)

    Article  Google Scholar 

  54. Yamamoto, T.: Error-bounds for Newton iterates derived from the Kantorovich-theorem. Numer. Math. 48, 91–98 (1986)

    Google Scholar 

  55. Yamamoto, T.: A method for finding sharp error-bounds for Newton method under the Kantorivich assumptions. Numer. Math. 49, 203–220 (1986)

    Google Scholar 

  56. Yamamoto, T.: Historical developments in convergence analysis for Newton’s and Newton-like methods. J. Comput. Appl. Math. 124 (1-2), 1–23 (2000), Numerical analysis 2000, Vol. IV, Optimization and nonlinear equations

  57. Ye, Y.: Interior Point Algorithms : Theory and Analysis. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley and Sons, 1997

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Correspondence to Florian A. Potra.

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Mathematics Subject Classification (2000): 49M15, 65H10, 65K05, 90C33

Work supported by the National Science Foundation under Grants No. 9996154, 0139701

Acknowledgement The original version of this paper [38] was written when the author was visiting the Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB). The author would like to thank Peter Deuflhard, President of ZIB, for the excellent working conditions at ZIB, and for very interesting discussions on various mathematical problems related to the subject of the present paper. The author would also like to thank two anonymous referees whose comments and suggestions led to a better presentation of our results.

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Potra, F. The Kantorovich Theorem and interior point methods. Math. Program. 102, 47–70 (2005). https://doi.org/10.1007/s10107-003-0501-8

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