Interior proximal method for variational inequalities on non-polyhedral sets

Alexander Kaplan; Rainer Tichatschke

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2007)

  • Volume: 27, Issue: 1, page 71-93
  • ISSN: 1509-9407

Abstract

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Interior proximal methods for variational inequalities are, in fact, designed to handle problems on polyhedral convex sets or balls, only. Using a slightly modified concept of Bregman functions, we suggest an interior proximal method for solving variational inequalities (with maximal monotone operators) on convex, in general non-polyhedral sets, including in particular the case in which the set is described by a system of linear as well as strictly convex constraints. The convergence analysis of the method studied admits the use of the 𝝐-enlargement of the operator and an inexact solution of the subproblems.

How to cite

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Alexander Kaplan, and Rainer Tichatschke. "Interior proximal method for variational inequalities on non-polyhedral sets." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 27.1 (2007): 71-93. <http://eudml.org/doc/271153>.

@article{AlexanderKaplan2007,
abstract = {Interior proximal methods for variational inequalities are, in fact, designed to handle problems on polyhedral convex sets or balls, only. Using a slightly modified concept of Bregman functions, we suggest an interior proximal method for solving variational inequalities (with maximal monotone operators) on convex, in general non-polyhedral sets, including in particular the case in which the set is described by a system of linear as well as strictly convex constraints. The convergence analysis of the method studied admits the use of the 𝝐-enlargement of the operator and an inexact solution of the subproblems.},
author = {Alexander Kaplan, Rainer Tichatschke},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {variational inequalities; Bregman function; proximal algorithm; Bergman function},
language = {eng},
number = {1},
pages = {71-93},
title = {Interior proximal method for variational inequalities on non-polyhedral sets},
url = {http://eudml.org/doc/271153},
volume = {27},
year = {2007},
}

TY - JOUR
AU - Alexander Kaplan
AU - Rainer Tichatschke
TI - Interior proximal method for variational inequalities on non-polyhedral sets
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2007
VL - 27
IS - 1
SP - 71
EP - 93
AB - Interior proximal methods for variational inequalities are, in fact, designed to handle problems on polyhedral convex sets or balls, only. Using a slightly modified concept of Bregman functions, we suggest an interior proximal method for solving variational inequalities (with maximal monotone operators) on convex, in general non-polyhedral sets, including in particular the case in which the set is described by a system of linear as well as strictly convex constraints. The convergence analysis of the method studied admits the use of the 𝝐-enlargement of the operator and an inexact solution of the subproblems.
LA - eng
KW - variational inequalities; Bregman function; proximal algorithm; Bergman function
UR - http://eudml.org/doc/271153
ER -

References

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  1. [1] A. Auslender and M. Haddou, An interior-proximal method for convex linearly constrained problems and its extension to variational inequalities, Math. Programming 71 (1995), 77-100. Zbl0855.90095
  2. [2] A. Auslender and M. Teboulle, Entropic proximal decomposition methods for convex programs and variational inequalities, Math. Programming Ser. A 91 (2001), 33-47. Zbl1051.90017
  3. [3] A. Auslender, M. Teboulle and S. Ben-Tiba, Interior proximal and multiplier methods based on second order homogeneous kernels, Mathematics of Oper. Res. 24 (1999), 645-668. Zbl1039.90518
  4. [4] A. Auslender, M. Teboulle and S. Ben-Tiba, A logarithmic-quadratic proximal method for variational inequalities, Computational Optimization and Applications 12 (1999), 31-40. 
  5. [5] H. Bauschke and J. Borwein, Legendre functions and the method of random Bregman projections, J. Convex Analysis 4 (1997), 27-67. Zbl0894.49019
  6. [6] H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier 18 (1968), 115-175. Zbl0169.18602
  7. [7] R. Burachik and A. Iusem, A generalized proximal point algorithm for the variational inequality problem in Hilbert space, SIAM J. Optim. 8 (1998), 197-216. Zbl0911.90273
  8. [8] R. Burachik, A. Iusem and B. Svaiter, Enlargements of maximal monotone operators with application to variational inequalities, Set-Valued Analysis 5 (1997), 159-180. Zbl0882.90105
  9. [9] R. Burachik and B. Svaiter, ϵ-enlargement of maximal monotone operators in Banach spaces, Set-Valued Analysis 7 (1999), 117-132. Zbl0948.47050
  10. [10] R. Burachik and B. Svaiter, A relative error tolerance for a family of generalized proximal point methods, Math. of Oper. Res. 26 (2001), 816-831. Zbl1082.65539
  11. [11] Y. Censor, A. Iusem and S.A. Zenios, An interior point method with Bregman functions for the variational inequality problem with paramonotone operators, Math. Programming 81 (1998), 373-400. Zbl0919.90123
  12. [12] Y. Censor and S.A. Zenios, Proximal minimization algorithm with d-functions, J. Optim. Theory Appl. 73 (1992), 451-464. Zbl0794.90058
  13. [13] J. Eckstein, Approximate iterations in Bregman-function-based proximal algorithms, Math. Programming 83 (1998), 113-123. Zbl0920.90117
  14. [14] A. Iusem, On some properties of generalized proximal point methods for quadratic and linear programming, JOTA 85 (1995), 593-612. Zbl0831.90092
  15. [15] A. Iusem, On some properties of paramonotone operators, J. of Conv. Analysis 5 (1998), 269-278. Zbl0914.90216
  16. [16] A. Kaplan and R. Tichatschke, Stable Methods for Ill-Posed Variational Problems - Prox-Regularization of Elliptic Variational Inequalities and Semi-Infinite Optimization Problems, Akademie Verlag, Berlin 1994. Zbl0804.49011
  17. [17] A. Kaplan and R. Tichatschke, Proximal point approach and approximation of variational inequalities, SIAM J. Control Optim. 39 (2000), 1136-1159. Zbl0997.47053
  18. [18] A. Kaplan and R. Tichatschke, Convergence analysis of non-quadratic proximal methods for variational inequalities in Hilbert spaces, J. of Global Optimization 22 (2002), 119-136. Zbl1047.49005
  19. [19] A. Kaplan and R. Tichatschke, Interior proximal method for variational inequalities: Case of non-paramonotone operators, Journal of Set-Valued Analysis 12 (2004), 357-382. Zbl1072.65093
  20. [20] A. Kaplan and R. Tichatschke, On inexact generalized proximal methods with a weakened error tolerance criterion, Optimization 53 (2004), 3-17. Zbl1068.65064
  21. [21] K. Kiwiel, Proximal minimization methods with generalized Bregman functions, SIAM J. Control Optim. 35 (1997), 1142-1168. Zbl0890.65061
  22. [22] J.L. Lions, Quelques Méthodes de Résolution de Problèmes Nonlinéaires, Dunod, Paris 1969. 
  23. [23] B. Martinet, Régularisation d'inéquations variationelles par approximations successives, RIRO 4 (1970), 154-159. 
  24. [24] N. Megiddo, Pathways to the optimal set in linear programming, In Progress in Mathematical Programming, Interior Point and Related Methods (1989), N. Megiddo, Ed., Springer, New York, pp. 131-158. 
  25. [25] D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Editura Academiei, Bucharest 1978. 
  26. [26] B.T. Polyak, Introduction to Optimization, Optimization Software, Inc. Publ. Division, New York 1987. Zbl0708.90083
  27. [27] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton 1970. Zbl0193.18401
  28. [28] R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75-88. Zbl0222.47017
  29. [29] R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), 877-898. Zbl0358.90053
  30. [30] M. Solodov and B. Svaiter, An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions, Math. Oper. Res. 25 (2000), 214-230. Zbl0980.90097
  31. [31] M. Teboulle, Convergence of proximal-like algorithms, SIAM J. Optim. 7 (1997), 1069-1083. Zbl0890.90151

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