Rescaled proximal methods for linearly constrained convex problems

Paulo J.S. Silva; Carlos Humes

RAIRO - Operations Research (2007)

  • Volume: 41, Issue: 4, page 367-380
  • ISSN: 0399-0559

Abstract

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We present an inexact interior point proximal method to solve linearly constrained convex problems. In fact, we derive a primal-dual algorithm to solve the KKT conditions of the optimization problem using a modified version of the rescaled proximal method. We also present a pure primal method. The proposed proximal method has as distinctive feature the possibility of allowing inexact inner steps even for Linear Programming. This is achieved by using an error criterion that bounds the subgradient of the regularized function, instead of using ϵ-subgradients of the original objective function. Quadratic convergence for LP is also proved using a more stringent error criterion.

How to cite

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Silva, Paulo J.S., and Humes, Carlos. "Rescaled proximal methods for linearly constrained convex problems." RAIRO - Operations Research 41.4 (2007): 367-380. <http://eudml.org/doc/250138>.

@article{Silva2007,
abstract = { We present an inexact interior point proximal method to solve linearly constrained convex problems. In fact, we derive a primal-dual algorithm to solve the KKT conditions of the optimization problem using a modified version of the rescaled proximal method. We also present a pure primal method. The proposed proximal method has as distinctive feature the possibility of allowing inexact inner steps even for Linear Programming. This is achieved by using an error criterion that bounds the subgradient of the regularized function, instead of using ϵ-subgradients of the original objective function. Quadratic convergence for LP is also proved using a more stringent error criterion. },
author = {Silva, Paulo J.S., Humes, Carlos},
journal = {RAIRO - Operations Research},
keywords = {Interior proximal methods; Linearly constrained convex problems; interior proximal methods; linearly constrained convex problems},
language = {eng},
month = {10},
number = {4},
pages = {367-380},
publisher = {EDP Sciences},
title = {Rescaled proximal methods for linearly constrained convex problems},
url = {http://eudml.org/doc/250138},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Silva, Paulo J.S.
AU - Humes, Carlos
TI - Rescaled proximal methods for linearly constrained convex problems
JO - RAIRO - Operations Research
DA - 2007/10//
PB - EDP Sciences
VL - 41
IS - 4
SP - 367
EP - 380
AB - We present an inexact interior point proximal method to solve linearly constrained convex problems. In fact, we derive a primal-dual algorithm to solve the KKT conditions of the optimization problem using a modified version of the rescaled proximal method. We also present a pure primal method. The proposed proximal method has as distinctive feature the possibility of allowing inexact inner steps even for Linear Programming. This is achieved by using an error criterion that bounds the subgradient of the regularized function, instead of using ϵ-subgradients of the original objective function. Quadratic convergence for LP is also proved using a more stringent error criterion.
LA - eng
KW - Interior proximal methods; Linearly constrained convex problems; interior proximal methods; linearly constrained convex problems
UR - http://eudml.org/doc/250138
ER -

References

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  1. A. Auslender and M. Haddou, An inteirior proximal method for convex linearly constrained problems. Math. Program.71 (1995) 77–100.  
  2. A. Auslender, M. Teboulle and S. Ben-Tiba, Interior proximal and multiplier methods based on second order homogeneous kernels. Math. Oper. Res.24 (1999) 645–668.  
  3. A. Auslender, M. Teboulle and S. Ben-Tiba, A logarithmic-quadratic proximal method for variational inequalities. Comput. Optim. Appl.12 (1999) 31–40.  
  4. Y. Censor and J. Zenios, The proximal minimization algorithms with D-functions. J. Optim. Theor. App.73 (1992) 451–464.  
  5. J. Eckstein, Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming. Math. Oper. Res.18 (1993) 202–226.  
  6. C.C. Gonzaga, Path-following methods for linear programming. SIAM Rev.34 (1992) 167–224.  
  7. A.N. Iusem and M. Teboulle, Convergence rate analysis of nonquadratic proximal methods for convex and linear programming. Math. Oper. Res.20 (1995) 657–677.  
  8. A.N. Iusem, M. Teboulle and B. Svaiter, Entropy-like proximal methods in covex programming. Math. Oper. Res.19 (1994) 790–814.  
  9. B. Martinet, Regularisation d'inequations variationelles par approximations successives. Rev. Fr. Inf. Rech. Oper. (1970) 154–159.  
  10. B. Martinet, Determination approch d'un point fixe d'une application pseudo-contractante. C.R. Acad. Sci. Paris274A (1972) 163–165.  
  11. R.T. Rockafellar, Convex Analysis. Princeton University Press (1970).  
  12. R.T. Rockafellar, Monotone operators and the proximal point algorithm. SIAM J. Control Optim.14 (1976) 887–898.  
  13. P.J.S. Silva and J. Eckstein, Double regularizations proximal methods, with complementarity applications. Comput. Optim. Appl.33 (2006) 115–116.  
  14. P.J.S. Silva, J. Eckstein and C. Humes Jr, Rescaling and stepsize selection in proximal methods using generalized distances. SIAM J. Optim.12 (2001) 238–261.  
  15. M. Teboulle, Entropic proximal methods with aplications to nonlinear programming. Math. Oper. Res.17 (1992) 670–690.  
  16. M. Teboulle, Convergence of proximal-like algorithms. SIAM J. Optim.7 (1997) 1069–1083.  
  17. P. Tseng and D. Bertesekas, On the convergence of the exponential multiplier method for convex programming. Math. Program.60 (1993) 1–19.  
  18. Stephen J. Wright, Primal-Dual Interior-Point Methods. SIAM (1997).  
  19. N. Yamashita, C. Kanzow, T. Morimoto and M. Fukushima, An infeasible interior proximal method for convex programming problems with linear constraints. Journal Nonlinear Convex Analysis2 (2001) 139–156.  

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