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Customized proximal point algorithms for linearly constrained convex minimization and saddle-point problems: a unified approach

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Abstract

This paper focuses on some customized applications of the proximal point algorithm (PPA) to two classes of problems: the convex minimization problem with linear constraints and a generic or separable objective function, and a saddle-point problem. We treat these two classes of problems uniformly by a mixed variational inequality, and show how the application of PPA with customized metric proximal parameters can yield favorable algorithms which are able to make use of the models’ structures effectively. Our customized PPA revisit turns out to unify some algorithms including some existing ones in the literature and some new ones to be proposed. From the PPA perspective, we establish the global convergence and a worst-case O(1/t) convergence rate for this series of algorithms in a unified way.

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References

  1. Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer., 1–137 (2005)

  2. Blum, E., Oettli, W.: Mathematische Optimierung. Grundlagen und Verfahren. Ökonometrie und Unternehmensforschung. Springer, Berlin (1975)

    Google Scholar 

  3. Bonnans, J.F., Gilbert, J.C., Lemarechal, C., Sagastizaba, C.A.: A family of variable-metric proximal methods. Math. Program. 68, 15–47 (1995)

    MATH  Google Scholar 

  4. Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3, 1–122 (2010)

    Article  MATH  Google Scholar 

  5. Burke, J.V., Qian, M.J.: A variable metric proximal point algorithm for monotone operators. SIAM J. Control Optim. 37, 353–375 (1998)

    Article  MathSciNet  Google Scholar 

  6. Burke, J.V., Qian, M.J.: On the superlinear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating. Math. Program. 88, 157–181 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cai, X.J., Gu, G., He, B.S., Yuan, X.M.: A proximal point algorithm revisit on alternating direction method of multipliers. Sci. China Math. 56(10), 2179–2186 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problem with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, S., Donoho, D., Saunders, M.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20, 33–61 (1998)

    Article  MathSciNet  Google Scholar 

  10. Condat, L.: A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. J. Optim. Theory Appl. (to appear)

  11. Douglas, J., Rachford, H.H.: On the numerical solution of the heat conduction problem in 2 and 3 space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Eckstein, J., Fukushima, M.: Some reformulation and applications of the alternating directions method of multipliers. In: Hager, W.W., et al. (eds.) Large Scale Optimization: State of the Art, pp. 115–134. Kluwer Academic, Dordrecht (1994)

    Chapter  Google Scholar 

  14. Esser, E., Zhang, X.Q., Chan, T.F.: A general framework for a class of first order primal-dual algorithms for convex pptimization in imaging science. SIAM J. Imaging Sci. 3, 1015–1046 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer Series in Operations Research. Springer, New York (2003)

    MATH  Google Scholar 

  16. Fortin, M., Brezzi, F.: Mixed and Hybrid Element Methods. Springer Series in Computational Mathematics. Springer, Berlin (1991)

    MATH  Google Scholar 

  17. Fukushima, M.: Application of the alternating directions method of multipliers to separable convex programming problems. Comput. Optim. Appl. 2, 93–111 (1992)

    Article  MathSciNet  Google Scholar 

  18. Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite-element approximations. Comput. Math. Appl. 2, 17–40 (1976)

    Article  MATH  Google Scholar 

  19. Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrange Methods: Applications to the Solution of Boundary-Valued Problems, pp. 299–331. North Holland, Amsterdam (1983)

    Google Scholar 

  20. Glowinski, R., Marrocco, A.: Approximation par éléments finis d’ordre un et résolution par pénalisation-dualité d’une classe de problémes non linéaires. RAIRO 2, 41–76 (1975)

    MathSciNet  Google Scholar 

  21. Gol’shtein, E.G., Tret’yakov, N.V.: Modified Lagrangian in convex programming and their generalizations. Math. Program. Stud. 10, 86–97 (1979)

    Article  MathSciNet  Google Scholar 

  22. Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ha, C.D.: A generalization of the proximal point algorithm. SIAM J. Control Optim. 28, 503–512 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  24. Han, D.R., Yuan, X.M., Zhang, W.X.: An augmented-Lagrangian-based parallel splitting method for separable convex programming with applications to image processing. Math. Comput. (to appear)

  25. Han, W.M., Reddy, B.D.: On the finite element method for mixed variational inequalities airising in elastoplasticity. SIAM J. Numer. Anal. 32, 1778–1807 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  26. He, B.S., Fu, X.L., Jiang, Z.K.: Proximal point algorithm using a linear proximal term. J. Optim. Theory Appl. 141, 299–319 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. He, B.S., Tao, M., Yuan, X.M.: A splitting method for separable convex programming. IMA J. Numer. Anal. (to appear)

  28. He, B.S., Tao, M., Yuan, X.M.: Alternating direction method with Gaussian back substitution for separable convex programming. SIAM J. Optim. 22, 313–340 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  29. He, B.S., Yang, H.: Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities. Oper. Res. Lett. 23, 151–161 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  30. He, B.S., Yuan, X.M., Zhang, W.X.: A customized proximal point algorithm for convex minimization with linear constraints. Comput. Optim. Appl. (to appear)

  31. He, B.S., Yuan, X.M.: Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective. SIAM J. Imaging Sci. 5, 119–149 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  32. He, B.S., Yuan, X.M.: On the O(1/n) convergence rate of Douglas-Rachford alternating direction method. SIAM J. Numer. Anal. 50, 700–709 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  33. Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 302–320 (1969)

    Google Scholar 

  34. Konnov, I.V.: Partial proximal point method for nonmonotone equilibrium problems. Optim. Methods Softw. 21, 373–384 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  35. Kontogiorgis, S., Meyer, R.R.: A variable-penalty alternating directions method for convex optimization. Math. Program. 83, 29–53 (1998)

    MATH  MathSciNet  Google Scholar 

  36. Lemaire, B.: Saddle point problems in partial differential equations and applications to linear quadratic differential games. Ann. Sc. Norm. Super. Pisa, Class Sci. 30 Sér. 27, 105–160 (1973)

  37. Martinet, B.: Regularisation, d’inéquations variationelles par approximations succesives. Rev. Fr. Inform. Rech. Oper. 4, 154–159 (1970)

    MATH  MathSciNet  Google Scholar 

  38. Medhi, D., Ha, C.D.: Generalized proximal point algorithms for convex optimization. J. Optim. Theory Appl. 88, 475–488 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  39. Monteiro, R.D.C., Svaiter, B.F.: Iteration-complexity of block-decomposition algorithms and the alternating direction method of multipliers. SIAM J. Optim. 23(1), 475–507 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  40. Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. Fr. 93, 273–299 (1965)

    MATH  MathSciNet  Google Scholar 

  41. Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2, 21–56 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  42. Pock, T., Chambolle, A.: Diagonal preconditioning for first order primal-dual algorithms in convex optimization. In: IEEE International Conference on Computer Vision (ICCV) (2011)

    Google Scholar 

  43. Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization. Academic Press, New York (1969)

    Google Scholar 

  44. Reddy, B.D.: Mixed variational inequalities arising in elastoplasticity. Nonlinear Anal. 19, 1071–1089 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  45. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  46. Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1, 97–116 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  47. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  48. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 227–238 (1992)

    Article  Google Scholar 

  49. Sun, J., Zhang, S.: A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs. Eur. J. Oper. Res. 207, 1210–1220 (2010)

    Article  MATH  Google Scholar 

  50. Vũ, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Manuscript (2011)

  51. Zhang, X.Q., Burger, M., Osher, S.: A unified primal-dual algorithm framework based on Bregman iteration. J. Sci. Comput. 46, 20–46 (2010)

    Article  MathSciNet  Google Scholar 

  52. Zhu, M., Chan, T.: An efficient primal-dual hybrid gradient algorithm for total variation image restoration. CAM Reports 08-34, UCLA, Center for Applied Mathematics (2008)

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Guoyong Gu

  2. International Center of Management Science and Engineering and Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Bingsheng He

  3. Department of Mathematics, Hong Kong Baptist University, Hong Kong, China

    Xiaoming Yuan

Authors
  1. Guoyong Gu
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  2. Bingsheng He
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  3. Xiaoming Yuan
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Corresponding author

Correspondence to Xiaoming Yuan.

Additional information

This paper is dedicated to Professor Masao Fukushima on the occasion of his 65th birthday.

G. Gu was supported by the NSFC grant 11001124. B. He was supported by the NSFC grant 91130007 and the MOEC fund 20110091110004. X. Yuan was supported by the General Research Fund from Hong Kong Research Grants Council: 203712.

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Gu, G., He, B. & Yuan, X. Customized proximal point algorithms for linearly constrained convex minimization and saddle-point problems: a unified approach. Comput Optim Appl 59, 135–161 (2014). https://doi.org/10.1007/s10589-013-9616-x

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