Mathematical Programming

, Volume 55, Issue 1–3, pp 293–318 | Cite as

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

  • Jonathan Eckstein
  • Dimitri P. Bertsekas
Article

Abstract

This paper shows, by means of an operator called asplitting operator, that the Douglas—Rachford splitting method for finding a zero of the sum of two monotone operators is a special case of the proximal point algorithm. Therefore, applications of Douglas—Rachford splitting, such as the alternating direction method of multipliers for convex programming decomposition, are also special cases of the proximal point algorithm. This observation allows the unification and generalization of a variety of convex programming algorithms. By introducing a modified version of the proximal point algorithm, we derive a new,generalized alternating direction method of multipliers for convex programming. Advances of this sort illustrate the power and generality gained by adopting monotone operator theory as a conceptual framework.

Key words

Monotone operators proximal point algorithm decomposition 

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References

  1. [1]
    D.P. Bertsekas, “Necessary and sufficient conditions for a penalty method to be exact,”Mathematical Programming 9 (1975) 87–99.Google Scholar
  2. [2]
    D.P. Bertsekas,Constrained Optimization and Lagrange Multiplier Methods (Academic Press, New York, 1982).Google Scholar
  3. [3]
    D.P. Bertsekas and J. Tsitsiklis,Parallel and Distributed Computation: Numerical Methods (Prentice-Hall, Englewood Cliffs, NJ, 1989).Google Scholar
  4. [4]
    H. Brézis,Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert (North-Holland, Amsterdam, 1973).Google Scholar
  5. [5]
    H. Brézis and P.-L. Lions, “Produits infinis de resolvantes,”Israel Journal of Mathematics 29 (1978) 329–345.Google Scholar
  6. [6]
    V. Doležal,Monotone Operators and Applications in Control and Network Theory (Elsevier, Amsterdam, 1979).Google Scholar
  7. [7]
    J. Douglas and H.H. Rachford, “On the numerical solution of heat conduction problems in two and three space variables,”Transactions of the American Mathematical Society 82 (1956) 421–439.Google Scholar
  8. [8]
    R. Durier, “On locally polyhedral convex functions,” Working paper, Université de Dijon (Dijon, 1986).Google Scholar
  9. [9]
    R. Durier and C. Michelot, “Sets of efficient points in a normed space,”Journal of Mathematical Analysis and its Applications 117 (1986) 506–528.Google Scholar
  10. [10]
    J. Eckstein, “The Lions—Mercier algorithm and the alternating direction method are instances of the proximal point algorithm,” Report LIDS-P-1769, Laboratory for Information and Decision Sciences, MIT (Cambridge, MA, 1988).Google Scholar
  11. [11]
    J. Eckstein, “Splitting methods for monotone operators with applications to parallel optimization,” Doctoral dissertation, Department of Civil Engineering, Massachusetts Institute of Technology. Available as Report LIDS-TH-1877, Laboratory for Information and Decision Sciences, MIT (Cambridge, MA, 1989).Google Scholar
  12. [12]
    M. Fortin and R. Glowinski, “On decomposition-coordination methods using an augmented Lagrangian,” in: M. Fortin and R. Glowinski, eds.,Augmented Lagrangian Methods: Applications to the Solution of Boundary-Value Problems (North-Holland, Amsterdam, 1983).Google Scholar
  13. [13]
    D. Gabay, “Applications of the method of multipliers to variational inequalities,” in: M. Fortin and R. Glowinski, eds.,Augmented Lagrangian Methods: Applications to the Solution of Boundary-Value Problems (North-Holland, Amsterdam, 1983).Google Scholar
  14. [14]
    D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximations,”Computers and Mathematics with Applications 2 (1976) 17–40.Google Scholar
  15. [15]
    R. Glowinski and P. Le Tallec, “Augmented Lagrangian methods for the solution of variational problems,” MRC Technical Summary Report #2965, Mathematics Research Center, University of Wisconsin-Madison (Madison, WI, 1987).Google Scholar
  16. [16]
    R. Glowinski and A. Marroco, “Sur l'approximation, par elements finis d'ordre un, et la resolution, par penalisation-dualité, d'une classe de problemes de Dirichlet non lineares,”Revue Française d'Automatique, Informatique et Recherche Opérationelle 9(R-2) (1975) 41–76.Google Scholar
  17. [17]
    E.G. Gol'shtein, “Method for modification of monotone mappings,”Ekonomika i Matemacheskaya Metody 11 (1975) 1142–1159.Google Scholar
  18. [18]
    E.G. Gol'shtein, “Decomposition methods for linear and convex programming problems,”Matekon 22(4) (1985) 75–101.Google Scholar
  19. [19]
    E.G. Gol'shtein, “The block method of convex programming,”Soviet Mathematics Doklady 33 (1986) 584–587.Google Scholar
  20. [20]
    E.G. Gol'shtein, “A general approach to decomposition of optimization systems,”Soviet Journal of Computer and Systems Sciences 25(3) (1987) 105–114.Google Scholar
  21. [21]
    E.G. Gol'shtein and N.V. Tret'yakov, “The gradient method of minimization and algorithms of convex programming based on modified Lagrangian functions,”Ekonomika i Matemacheskaya Metody 11(4) (1975) 730–742.Google Scholar
  22. [22]
    E.G. Gol'shtein and N.V. Tret'yakov, “Modified Lagrangians in convex programming and their generalizations,”Mathematical Programming Study 10 (1979) 86–97.Google Scholar
  23. [23]
    M.R. Hestenes, “Multiplier and gradient methods,”Journal of Optimization Theory and Applications 4 (1969) 303–320.Google Scholar
  24. [24]
    M.C. Joshi and R.K. Bose,Some Topics in Nonlinear Functional Analysis (Halsted/Wiley, New Delhi, 1985).Google Scholar
  25. [25]
    R.I. Kachurovskii, “On monotone operators and convex functionals,”Uspekhi Matemacheskaya Nauk 15(4) (1960) 213–215.Google Scholar
  26. [26]
    R.I. Kachurovskii, “Nonlinear monotone operators in Banach space,”Russian Mathematical Surveys 23(2) (1968) 117–165.Google Scholar
  27. [27]
    B. Kort and D.P. Bertsekas, “Combined primal—dual and penalty methods for convex programming,”SIAM Journal on Control and Optimization 14 (1976) 268–294.Google Scholar
  28. [28]
    J. Lawrence and J.E. Spingarn, “On fixed points of non-expansive piecewise isometric mappings,”Proceedings of the London Mathematical Society 55 (1987) 605–624.Google Scholar
  29. [29]
    O. Lefebvre and C. Michelot, “About the finite convergence of the proximal point algorithm,” in: K.-H. Hoffmann et al., eds.,Trends in Mathematical Optimization: 4th French—German Conference on Optimization. International Series of Numerical Mathematics No. 84 (Birkhäuser, Basel, 1988).Google Scholar
  30. [30]
    P.-L. Lions, “Une méthode itérative de resolution d'une inequation variationnelle,”Israel Journal of Mathematics 31 (1978) 204–208.Google Scholar
  31. [31]
    P.-L. Lions and B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,”SIAM Journal on Numerical Analysis 16 (1979) 964–979.Google Scholar
  32. [32]
    F.J. Luque, “Asymptotic convergence analysis of the proximal point algorithm,”SIAM Journal on Control and Optimization 22 (1984) 277–293.Google Scholar
  33. [33]
    G.I. Marchuk,Methods of Numerical Mathematics (Springer, New York, 1975).Google Scholar
  34. [34]
    B. Martinet, “Regularisation d'inequations variationelles par approximations successives,”Revue Française d'Informatique et de Recherche Operationelle 4(R-3) (1970) 154–158.Google Scholar
  35. [35]
    B. Martinet, “Determination approchée d'un point fixe d'une application pseudo-contractante. Cas de l'application prox,”Comptes Rendus de l'Academie des Sciences, Paris, Série A 274 (1972) 163–165.Google Scholar
  36. [36]
    G.J. Minty, “On the maximal domain of a ‘monotone’ function,”Michigan Mathematical Journal 8 (1961) 135–137.Google Scholar
  37. [37]
    G.J. Minty, “Monotone (nonlinear) operators in Hilbert space,”Duke Mathematics Journal 29 (1962) 341–346.Google Scholar
  38. [38]
    G.J. Minty, “On the monotonicity of the gradient of a convex function,”Pacific Journal of Mathematics 14 (1964) 243–247.Google Scholar
  39. [39]
    D. Pascali and S. Sburlan,Nonlinear Mappings of Monotone Type (Editura Academeie, Bucarest, 1978).Google Scholar
  40. [40]
    G.B. Passty, “Ergodic convergence to a zero of the sum of monotone operators in Hilbert space,”Journal of Mathematical Analysis and Applications 72 (1979) 383–390.Google Scholar
  41. [41]
    M.J.D. Powell, “A method for nonlinear constraints in minimization problems,” in: R. Fletcher, ed.,Optimization (Academic Press, New York, 1969).Google Scholar
  42. [42]
    R.T. Rockafellar, “Characterization of the subdifferentials of convex functions,”Pacific Journal of Mathematics 17 (1966) 497–510.Google Scholar
  43. [43]
    R.T. Rockafellar, “Local boundedness of nonlinear, monotone operators,”Michigan Mathematical Journal 16 (1969) 397–407.Google Scholar
  44. [44]
    R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).Google Scholar
  45. [45]
    R.T. Rockafellar, “On the maximal monotonicity of subdifferential mappings,”Pacific Journal of Mathematics 33 (1970) 209–216.Google Scholar
  46. [46]
    R.T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,”Transactions of the American Mathematical Society 149 (1970) 75–88.Google Scholar
  47. [47]
    R.T. Rockafellar, “On the virtual convexity of the domain and range of a nonlinear maximal monotone operator,”Mathematische Annalen 185 (1970) 81–90.Google Scholar
  48. [48]
    R.T. Rockafellar, “Monotone operators and the proximal point algorithm,”SIAM Journal on Control and Optimization 14 (1976) 877–898.Google Scholar
  49. [49]
    R.T. Rockafellar, “Augmented Lagrangians and applications of the proximal point algorithm in convex programming,”Mathematics of Operations Research 1 (1976) 97–116.Google Scholar
  50. [50]
    R.T. Rockafellar, “Monotone operators and augmented Lagrangian methods in nonlinear programming,” in: O.L. Mangasarian, R.M. Meyer and S.M. Robinson, eds.,Nonlinear Programming 3 (Academic Press, New York, 1977).Google Scholar
  51. [51]
    R.T. Rockafellar and R.J.-B. Wets, “Scenarios and policy aggregation in optimization under uncertainty,”Mathematics of Operations Research 16(1) (1991) 119–147.Google Scholar
  52. [52]
    J.E. Spingarn, “Partial inverse of a monotone operator,”Applied Mathematics and Optimization 10 (1983) 247–265.Google Scholar
  53. [53]
    J.E. Spingarn, “A primal—dual projection method for solving systems of linear inequalities,”Linear Algebra and its Applications 65 (1985) 45–62.Google Scholar
  54. [54]
    J.E. Spingarn, “Application of the method of partial inverses to convex programming: decomposition,”Mathematical Programming 32 (1985) 199–233.Google Scholar
  55. [55]
    J.E. Spingarn, “A projection method for least-squares solutions to overdetermined systems of linear inequalities,”Linear Algebra and its Applications 86 (1987) 211–236.Google Scholar
  56. [56]
    P. Tseng, “Applications of a splitting algorithm to decomposition in convex programming and variational inequalities,”SIAM Journal on Control and Optimization 29(1) (1991) 119–138.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1992

Authors and Affiliations

  • Jonathan Eckstein
    • 1
  • Dimitri P. Bertsekas
    • 2
  1. 1.Mathematical Sciences Research GroupThinking Machines CorporationCambridgeUSA
  2. 2.Laboratory for Information and Decision SystemsMassachusetts Institute of TechnologyCambridgeUSA

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