Skip to main content

Advertisement

SpringerLink
Log in

Solving variational inequality and fixed point problems by line searches and potential optimization

  • Published:
Mathematical Programming Submit manuscript

Abstract.

We introduce a general adaptive line search framework for solving fixed point and variational inequality problems. Our goals are to develop iterative schemes that (i) compute solutions when the underlying map satisfies properties weaker than contractiveness, for example, weaker forms of nonexpansiveness, (ii) are more efficient than the classical methods even when the underlying map is contractive, and (iii) unify and extend several convergence results from the fixed point and variational inequality literatures. To achieve these goals, we introduce and study joint compatibility conditions imposed upon the underlying map and the iterative step sizes at each iteration and consider line searches that optimize certain potential functions. As a special case, we introduce a modified steepest descent method for solving systems of equations that does not require a previous condition from the literature (the square of the Jacobian matrix is positive definite). Since the line searches we propose might be difficult to perform exactly, we also consider inexact line searches.

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. Baillon, J.B.: Un theoreme de type ergodique pour les contractions non lineaires dans un espace de Hilbert. C.R. Acad. Sc., Serie A, Paris t.28, 1511–1514 (1975)

  2. Bertsekas, D.P.: Nonlinear programming. Athena Scientific, Belmont, MA (1996)

  3. Bottom, J.(1998): Generation of Anticipatory Route Guidance, presentation in TRISTAN III, Triennial Symposium on Transportation Analysis, San Juan, Puerto Rico, also in Optimization Days, Montreal, May 1999

  4. Bottom, J., Ben-Akiva, M., Bierlaire, M., Chabini, I., Koutsopoulos, H., Yang, Q.: Investigation of route guidance generation issues by simulation with DynaMIT”, to appear in Proceedings of 14th International Symposium on Transportation and Traffic Theory, Jerusalem (1999)

  5. Browder, F.E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Rational Mech. Anal. 24, 82–90 (1967)

    MATH  Google Scholar 

  6. Browder, F.E., Petryshn, W.V.: Construction of fixed points of nonlinear mappings in a Hilbert space. Journal of Analysis and Applications 20, 197–228 (1967)

    MATH  Google Scholar 

  7. Bruck, Jr. R.E.: The iterative solution of the equation y ∈x+T(x) for a maximal monotone map T in Hilbert space. Journal of Mathematical Analysis and Applications 48, 114–126 (1973)

    MATH  Google Scholar 

  8. Bruck, Jr. R.E. : On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. Journal of Mathematical Analysis and Applications 61, 159–164 (1977)

    Article  MATH  Google Scholar 

  9. Calamai, P.H., More, J.J.: Projected gradient methods for linearly constrained problems. Mathematical Programming 39, 93–116 (1987)

    MathSciNet  MATH  Google Scholar 

  10. Christodouleas, J.: Applications of fixed point problems in scheduling. Private communication (1996)

  11. Dafermos, S.C.: An iterative scheme for variational inequalities. Mathematical Programming 26, 40–47 (1983)

    MathSciNet  MATH  Google Scholar 

  12. Dunn, J.C.: On recursive averaging processes and Hilbert space extensions of the contraction mapping principle. Journal of the Franklin Institute 295, 117–133 (1973)

    Article  MATH  Google Scholar 

  13. Dunn, J.C.: Convergence rates for conditional gradient sequences generated by implicit step length rules. SIAM Journal on Control and Optimization 18 (5), 473–487 (1980)

    MATH  Google Scholar 

  14. Dunn, J.C.: Newton’s method and the Goldstein step length rue for constrained minimization problems. SIAM Journal on Control and Optimization 18, 659–674 (1980)

    MathSciNet  MATH  Google Scholar 

  15. Dunn, J.C.: Global and asymptotic convergence rate estimates for a class of projected gradient processes. SIAM Journal on Control and Optimization 19 (3), 368–400 (1981)

    MATH  Google Scholar 

  16. Dunn, J.C., Harshbarger, S.: Conditional gradient algorithms with open loop step size rules. Journal of Mathematical Analysis and Applications 62, 432–444 (1978)

    Article  MATH  Google Scholar 

  17. Eckstein, J., Bertsekas, D. P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming 55, 293–318 (1992)

    MathSciNet  MATH  Google Scholar 

  18. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Volumes I and II, Springer Verlag, Berlin (2003)

  19. Florian, M., Hearn, D.: Network Equilibria. Handbook of Operations Research and Management Science. Volume Network Routing (M. Ball, T. Magnanti, C. Monma and G. Nemhauser eds.), Elsevier (1995)

  20. M. Fukushima: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Mathematical Programming 53 (1), 99–110 (1992)

    Google Scholar 

  21. Gabay, D.: Applications of the method of multipliers to variational inequalities, in M. Fortin, R. Glowinski, eds., Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, North-Holland, Amsterdam (1983)

  22. Halpern, B.: Fixed points of nonexpansive maps. Bulletin of the American Mathematical Society 73, 957–961 (1967)

    MATH  Google Scholar 

  23. Hammond, J.H., Magnanti, T.L.: Generalized descent methods for asymmetric systems of equations. Mathematics of Operations Research 12, 678–699 (1987)

    MathSciNet  MATH  Google Scholar 

  24. Harker, P.T.: Lectures on computation of equilibria with equation-based methods. CORE Lecture Series, Center for Operations Research and Econometrics, University of Louvain, Louvain la Neuve, Belgium (1993)

  25. Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Mathematical Programming 48, 161–220 (1990)

    MathSciNet  MATH  Google Scholar 

  26. Hoang, T.: Computing fixed points by global optimization methods, in Fixed point theory and applications M.A. Thera and J-B Baillon Eds. 231–244 (1991)

  27. Ishikawa, S.: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proceedings of the American Mathematical Society 59 (1), 65–71 (1976)

    MATH  Google Scholar 

  28. Itoh, T., Fukushima, M., Ibaraki, T.: An iterative method for variational inequalities with application to traffic equilibrium problems. Journal of the Operations Research Society of Japan 31 (1), 82–103 (1988)

    MATH  Google Scholar 

  29. Kaniel, S.: Construction of a fixed point for contractions in banach space. Israel J. Math. 9, 535–540 (1971)

    MATH  Google Scholar 

  30. Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their application. Academic Press, New York (1980)

  31. Knopp, K.: Theory and application of infinite series. Hafner Publishing Company N.Y. (1947)

  32. Kolmogorov, A.N., Fomin, S.V.: Introduction to real analysis. Dover Publications, New York (1970)

  33. Magnanti, T.L., Perakis, G.: A unifying geometric solution framework and complexity analysis for variational inequalities. Mathematical Programming 71, 327–351 (1995)

    Article  MathSciNet  Google Scholar 

  34. Magnanti, T.L., Perakis, G.: The orthogonality theorem and the strong-f-monotonicity condition for variational inequality algorithms. SIAM Journal of Optimization 7 (1), 248–273 (1997)

    Article  MATH  Google Scholar 

  35. Magnanti, T.L., Perakis, G.: Averaging schemes for variational inequalities and systems of equations. Mathematics of Operations Research 22 (3), 568–587 (1997)

    MATH  Google Scholar 

  36. Mann, W.R.: On recursive averaging processes and Hilbert space extensions of a contraction mapping principle. Proceedings of the American Mathematical Society 4, 506–510 (1953)

    MathSciNet  MATH  Google Scholar 

  37. Marcotte, P., Dussault, J-P: A note on a globally convergent Newton method for solving monotone variational inequalities. Operations Research Letters 6 (1), 35–42 (1987)

    Article  MATH  Google Scholar 

  38. Marcotte, P., Wu, J.H.: On the convergence of projection methods: application to the decomposition of affine variational inequalities. JOTA 85 (2), 347–362 (1995)

    MATH  Google Scholar 

  39. Nagurney, A.: Network Economics. Kluwer Academic Publishers, Boston (1993)

  40. Opial, Z.: Weak convergence of the successive approximation for non-expansive mappings in Banach Spaces. Bulletin A.M.S. 73, 123–135 (1967)

    Google Scholar 

  41. Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. Academic Press , New York (1970)

  42. Pang, J.S., Chan, D.: Iterative methods for variational and complementarity problems. Mathematical Programming 24, 284–313 (1982)

    MathSciNet  MATH  Google Scholar 

  43. Pang, J.S.: Complementarity problems. Handbook of Global Optimization, eds. R. Horst and P. Pardalos, 271–329, Kluwer Academic Publishers. (1994)

  44. Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. Journal of Mathematical Analysis and Applications 72, 383–390 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  45. Rockafellar, R.T.: Monotone operators and the proximal algorithm. SIAM Journal on Control and Optimization 14, 877–898 (1976)

    MATH  Google Scholar 

  46. Sibony, M.: Methodes iteratives pour les equations et inequations aux derivees partielles nonlinearies de type monotone. Calcolo 7, 65–183 (1970)

    MATH  Google Scholar 

  47. Sine, R.C.: Fixed Points and Nonexpansive Mappings. Contemporary Mathematics, 18, American Mathematical Society (1983)

  48. Sun, J.: An affine scaling method for linearly constrained convex programs. Preprint, Department of Industrial Engineering and Management Sciences, Northwestern University (1990)

  49. Taji, K., Fukushima, M., Ibaraki, T.: A globally convergent Newton method for solving strongly monotone variational inequalities. Mathematical Programming 58 (3), 369–383 (1993)

    MATH  Google Scholar 

  50. Todd, M.J.: The computation of fixed points and applications. Lecture Notes in Economics and Mathematical Systems 124, Springer-Verlag (1976)

  51. Tseng, P.: Further applications of a matrix splitting algorithm to decomposition in variational inequalities and convex programming. Mathematical Programming 48, 249–264 (1990)

    MathSciNet  MATH  Google Scholar 

  52. Tseng, P., Bertsekas, D.P., Tsitsiklis, J.N.: Partially asynchronous, parallel algorithms for network flow problems and other problems. SIAM Journal on Control and Optimization 28 (3), 678–710 (1990)

    MATH  Google Scholar 

  53. Wu, J.H., Florian, M., Marcotte, P.: A general descent framework for the monotone variational inequality problem. Mathematical Programming 61 (3), 281–300 (1993)

    MATH  Google Scholar 

  54. Zhu, D.L., Marcotte, P.: Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM Journal on Optimization 3 (6), 714–726(1996)

    MATH  Google Scholar 

  55. Zhu, D.L., Marcotte, P.: Modified descent methods for solving the monotone variational inequality problem. Operations Research Letters 14 (2), 111–120 (1993)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Engineering and Sloan School of Management MIT, Cambridge, MA, 02139

    Thomas L. Magnanti

  2. Sloan School of Management and Operations Research Center, MIT, Cambridge, MA, 02139

    Georgia Perakis

Authors
  1. Thomas L. Magnanti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Georgia Perakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Preparation of this paper was supported, in part, from the National Science Foundation NSF Grant 9634736-DMI, as well as the Singapore-MIT Alliance

Acknowledgments.We are grateful to the associate editor and the referees for their insightful comments and suggestions that have helped us improve both the exposition and the content of this paper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Magnanti, T., Perakis, G. Solving variational inequality and fixed point problems by line searches and potential optimization. Math. Program., Ser. A 101, 435–461 (2004). https://doi.org/10.1007/s10107-003-0476-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-003-0476-5

Keywords

Search

Navigation