On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
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.
- D.P. Bertsekas, “Necessary and sufficient conditions for a penalty method to be exact,”Mathematical Programming 9 (1975) 87–99.
- D.P. Bertsekas,Constrained Optimization and Lagrange Multiplier Methods (Academic Press, New York, 1982).
- D.P. Bertsekas and J. Tsitsiklis,Parallel and Distributed Computation: Numerical Methods (Prentice-Hall, Englewood Cliffs, NJ, 1989).
- H. Brézis,Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert (North-Holland, Amsterdam, 1973).
- H. Brézis and P.-L. Lions, “Produits infinis de resolvantes,”Israel Journal of Mathematics 29 (1978) 329–345.
- V. Doležal,Monotone Operators and Applications in Control and Network Theory (Elsevier, Amsterdam, 1979).
- 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.
- R. Durier, “On locally polyhedral convex functions,” Working paper, Université de Dijon (Dijon, 1986).
- R. Durier and C. Michelot, “Sets of efficient points in a normed space,”Journal of Mathematical Analysis and its Applications 117 (1986) 506–528.
- 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).
- 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).
- 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).
- 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).
- 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.
- 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).
- 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.
- E.G. Gol'shtein, “Method for modification of monotone mappings,”Ekonomika i Matemacheskaya Metody 11 (1975) 1142–1159.
- E.G. Gol'shtein, “Decomposition methods for linear and convex programming problems,”Matekon 22(4) (1985) 75–101.
- E.G. Gol'shtein, “The block method of convex programming,”Soviet Mathematics Doklady 33 (1986) 584–587.
- E.G. Gol'shtein, “A general approach to decomposition of optimization systems,”Soviet Journal of Computer and Systems Sciences 25(3) (1987) 105–114.
- 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.
- E.G. Gol'shtein and N.V. Tret'yakov, “Modified Lagrangians in convex programming and their generalizations,”Mathematical Programming Study 10 (1979) 86–97.
- M.R. Hestenes, “Multiplier and gradient methods,”Journal of Optimization Theory and Applications 4 (1969) 303–320.
- M.C. Joshi and R.K. Bose,Some Topics in Nonlinear Functional Analysis (Halsted/Wiley, New Delhi, 1985).
- R.I. Kachurovskii, “On monotone operators and convex functionals,”Uspekhi Matemacheskaya Nauk 15(4) (1960) 213–215.
- R.I. Kachurovskii, “Nonlinear monotone operators in Banach space,”Russian Mathematical Surveys 23(2) (1968) 117–165.
- 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.
- 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.
- 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).
- P.-L. Lions, “Une méthode itérative de resolution d'une inequation variationnelle,”Israel Journal of Mathematics 31 (1978) 204–208.
- P.-L. Lions and B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,”SIAM Journal on Numerical Analysis 16 (1979) 964–979.
- F.J. Luque, “Asymptotic convergence analysis of the proximal point algorithm,”SIAM Journal on Control and Optimization 22 (1984) 277–293.
- G.I. Marchuk,Methods of Numerical Mathematics (Springer, New York, 1975).
- B. Martinet, “Regularisation d'inequations variationelles par approximations successives,”Revue Française d'Informatique et de Recherche Operationelle 4(R-3) (1970) 154–158.
- 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.
- G.J. Minty, “On the maximal domain of a ‘monotone’ function,”Michigan Mathematical Journal 8 (1961) 135–137.
- G.J. Minty, “Monotone (nonlinear) operators in Hilbert space,”Duke Mathematics Journal 29 (1962) 341–346.
- G.J. Minty, “On the monotonicity of the gradient of a convex function,”Pacific Journal of Mathematics 14 (1964) 243–247.
- D. Pascali and S. Sburlan,Nonlinear Mappings of Monotone Type (Editura Academeie, Bucarest, 1978).
- 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.
- M.J.D. Powell, “A method for nonlinear constraints in minimization problems,” in: R. Fletcher, ed.,Optimization (Academic Press, New York, 1969).
- R.T. Rockafellar, “Characterization of the subdifferentials of convex functions,”Pacific Journal of Mathematics 17 (1966) 497–510.
- R.T. Rockafellar, “Local boundedness of nonlinear, monotone operators,”Michigan Mathematical Journal 16 (1969) 397–407.
- R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
- R.T. Rockafellar, “On the maximal monotonicity of subdifferential mappings,”Pacific Journal of Mathematics 33 (1970) 209–216.
- R.T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,”Transactions of the American Mathematical Society 149 (1970) 75–88.
- R.T. Rockafellar, “On the virtual convexity of the domain and range of a nonlinear maximal monotone operator,”Mathematische Annalen 185 (1970) 81–90.
- R.T. Rockafellar, “Monotone operators and the proximal point algorithm,”SIAM Journal on Control and Optimization 14 (1976) 877–898.
- R.T. Rockafellar, “Augmented Lagrangians and applications of the proximal point algorithm in convex programming,”Mathematics of Operations Research 1 (1976) 97–116.
- 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).
- 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.
- J.E. Spingarn, “Partial inverse of a monotone operator,”Applied Mathematics and Optimization 10 (1983) 247–265.
- J.E. Spingarn, “A primal—dual projection method for solving systems of linear inequalities,”Linear Algebra and its Applications 65 (1985) 45–62.
- J.E. Spingarn, “Application of the method of partial inverses to convex programming: decomposition,”Mathematical Programming 32 (1985) 199–233.
- 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.
- 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.
- On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators
Volume 55, Issue 1-3 , pp 293-318
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Monotone operators
- proximal point algorithm
- Industry Sectors