Abstract and Applied Analysis

A New Implementable Prediction-Correction Method for Monotone Variational Inequalities with Separable Structure

Feng Ma, Mingfang Ni, and Zhanke Yu

Full-text: Open access

Abstract

The monotone variational inequalities capture various concrete applications arising in many areas. In this paper, we develop a new prediction-correction method for monotone variational inequalities with separable structure. The new method can be easily implementable, and the main computational effort in each iteration of the method is to evaluate the proximal mappings of the involved operators. At each iteration, the algorithm also allows the involved subvariational inequalities to be solved in parallel. We establish the global convergence of the proposed method. Preliminary numerical results show that the new method can be competitive with Chen's proximal-based decomposition method in Chen and Teboulle (1994).

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 941861, 8 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393450487

Digital Object Identifier
doi:10.1155/2013/941861

Mathematical Reviews number (MathSciNet)
MR3111803

Zentralblatt MATH identifier
07095519

Citation

Ma, Feng; Ni, Mingfang; Yu, Zhanke. A New Implementable Prediction-Correction Method for Monotone Variational Inequalities with Separable Structure. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 941861, 8 pages. doi:10.1155/2013/941861. https://projecteuclid.org/euclid.aaa/1393450487


Export citation

References

  • S. Dafermos, “Traffic equilibrium and variational inequalities,” Transportation Science, vol. 14, no. 1, pp. 42–54, 1980.
  • D. P. Bertsekas and E. M. Gafni, Projection Methods for Variational Inequalities with Application to the Traffic Assignment Problem, Springer, Berlin, Germany, 1982.
  • B. Martinet, “Régularisation d'inéquations variationnelles par approximations successives,” Revue Française dInformatique et de Recherche Opérationelle, vol. 4, pp. 154–158, 1970.
  • R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976.
  • R. T. Rockafellar, “Augmented lagrangians and applications of the proximal point algorithm in convex programming,” Mathematics of Operations Research, vol. 1, no. 2, pp. 97–116, 1976.
  • B. He, L. Liao, and M. Qian, “Alternating projection based prediction-correction methods for structured variational inequalities,” Journal of Computational Mathematics, vol. 24, no. 6, pp. 693–710, 2006.
  • P. Tseng, “Alternating projection-proximal methods for convex programming and variational inequalities,” SIAM Journal on Optimization, vol. 7, no. 4, pp. 951–965, 1997.
  • B. He, L. Liao, D. Han, and H. Yang, “A new inexact alternating directions method for monotone variational inequalities,” Mathematical Programming B, vol. 92, no. 1, pp. 103–118, 2002.
  • B. He, S. Wang, and H. Yang, “A modified variable-penalty alternating directions method for monotone variational inequalities,” Journal of Computational Mathematics, vol. 21, pp. 495–504, 2003.
  • B. He, H. Yang, and S. Wang, “Alternating direction method with self-adaptive penalty parameters for monotone variational inequalities,” Journal of Optimization Theory and Applications, vol. 106, no. 2, pp. 337–356, 2000.
  • B. He, L. Liao, and X. Wang, “Proximal-like contraction methods for monotone variational inequalities in a unified framework II: general methods and numerical experiments,” Computational Optimization and Applications, vol. 51, no. 2, pp. 681–708, 2012.
  • B. He, Z. Yang, and X. Yuan, “An approximate proximal-extragradient type method for monotone variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 300, no. 2, pp. 362–374, 2004.
  • D. Han, “A hybrid entropic proximal decomposition method with self-adaptive strategy for solving variational inequality problems,” Computers and Mathematics with Applications, vol. 55, no. 1, pp. 101–115, 2008.
  • B. He, “Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities,” Computational Optimization and Applications, vol. 42, no. 2, pp. 195–212, 2009.
  • M. Xu, J. Jiang, B. Li, and B. Xu, “An improved prediction-correction method for monotone variational inequalities with separable operators,” Computers and Mathematics with Applications, vol. 59, no. 6, pp. 2074–2086, 2010.
  • M. Xu and T. Wu, “A class of linearized proximal alternating direction methods,” Journal of Optimization Theory and Applications, vol. 151, no. 2, pp. 321–337, 2011.
  • X. Yuan and M. Li, “An LQP-based decomposition method for solving a class of variational inequalities,” SIAM Journal on Optimization, vol. 21, no. 4, pp. 1309–1318, 2011.
  • A. Bnouhachem, H. Benazza, and M. Khalfaoui, “An inexact alternating direction method for solving a class of structured variational inequalities,” Applied Mathematics and Computation, vol. 219, no. 14, pp. 7837–7846, 2013.
  • B. He and X. Yuan, “Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective,” SIAM Journal on Imaging Sciences, vol. 5, no. 1, pp. 119–149, 2012.
  • D. Han, “A generalized proximal-point-based prediction-correction method for variational inequality problems,” Journal of Computational and Applied Mathematics, vol. 221, no. 1, pp. 183–193, 2008.
  • B. He and X. Yuan, “The unified framework of some proximal-based decomposition methods for monotone variational inequalities with separable structure,” Pacific Journal of Optimization, vol. 8, pp. 817–844, 2013.
  • G. Chen and M. Teboulle, “A proximal-based decomposition method for convex minimization problems,” Mathematical Programming, vol. 64, no. 1–3, pp. 81–101, 1994.