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Interior Proximal Algorithm for Quasiconvex Programming Problems and Variational Inequalities with Linear Constraints

Author

Listed:
  • Arnaldo S. Brito

    (Federal University of Rio de Janeiro)

  • J. X. Cruz Neto

    (Federal University of Piaui)

  • Jurandir O. Lopes

    (Federal University of Piaui)

  • P. Roberto Oliveira

    (COPPE/Sistemas-Universidade Federal do Rio de Janeiro)

Abstract

In this paper, we propose two interior proximal algorithms inspired by the logarithmic-quadratic proximal method. The first method we propose is for general linearly constrained quasiconvex minimization problems. For this method, we prove global convergence when the regularization parameters go to zero. The latter assumption can be dropped when the function is assumed to be pseudoconvex. We also obtain convergence results for quasimonotone variational inequalities, which are more general than monotone ones.

Suggested Citation

  • Arnaldo S. Brito & J. X. Cruz Neto & Jurandir O. Lopes & P. Roberto Oliveira, 2012. "Interior Proximal Algorithm for Quasiconvex Programming Problems and Variational Inequalities with Linear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 217-234, July.
    • Handle: RePEc:spr:joptap:v:154:y:2012:i:1:d:10.1007_s10957-012-0002-0
    DOI: 10.1007/s10957-012-0002-0
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    References listed on IDEAS

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    1. J.P. Penot, 2003. "Characterization of Solution Sets of Quasiconvex Programs," Journal of Optimization Theory and Applications, Springer, vol. 117(3), pages 627-636, June.
    2. A. Daniilidis & N. Hadjisavvas, 1999. "Characterization of Nonsmooth Semistrictly Quasiconvex and Strictly Quasiconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 102(3), pages 525-536, September.
    3. Alfred Auslender & Marc Teboulle & Sami Ben-Tiba, 1999. "Interior Proximal and Multiplier Methods Based on Second Order Homogeneous Kernels," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 645-668, August.
    4. Souza, Sissy da S. & Oliveira, P.R. & da Cruz Neto, J.X. & Soubeyran, A., 2010. "A proximal method with separable Bregman distances for quasiconvex minimization over the nonnegative orthant," European Journal of Operational Research, Elsevier, vol. 201(2), pages 365-376, March.
    5. Alfredo N. Iusem & B. F. Svaiter & Marc Teboulle, 1994. "Entropy-Like Proximal Methods in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 19(4), pages 790-814, November.
    6. N. El Farouq, 2001. "Pseudomonotone Variational Inequalities: Convergence of the Auxiliary Problem Method," Journal of Optimization Theory and Applications, Springer, vol. 111(2), pages 305-322, November.
    7. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    8. Marc Teboulle, 1992. "Entropic Proximal Mappings with Applications to Nonlinear Programming," Mathematics of Operations Research, INFORMS, vol. 17(3), pages 670-690, August.
    9. N. El Farouq, 2001. "Pseudomonotone Variational Inequalities: Convergence of Proximal Methods," Journal of Optimization Theory and Applications, Springer, vol. 109(2), pages 311-326, May.
    10. A. Auslender & M. Teboulle, 2004. "Interior Gradient and Epsilon-Subgradient Descent Methods for Constrained Convex Minimization," Mathematics of Operations Research, INFORMS, vol. 29(1), pages 1-26, February.
    11. D. Aussel, 1998. "Subdifferential Properties of Quasiconvex and Pseudoconvex Functions: Unified Approach," Journal of Optimization Theory and Applications, Springer, vol. 97(1), pages 29-45, April.
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    Cited by:

    1. F. Lara, 2022. "On Strongly Quasiconvex Functions: Existence Results and Proximal Point Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 891-911, March.
    2. E. A. Papa Quiroz & S. Cruzado, 2022. "An inexact scalarization proximal point method for multiobjective quasiconvex minimization," Annals of Operations Research, Springer, vol. 316(2), pages 1445-1470, September.
    3. Minglu Ye & Yiran He, 2015. "A double projection method for solving variational inequalities without monotonicity," Computational Optimization and Applications, Springer, vol. 60(1), pages 141-150, January.
    4. Regina S. Burachik & Yaohua Hu & Xiaoqi Yang, 2022. "Interior quasi-subgradient method with non-Euclidean distances for constrained quasi-convex optimization problems in hilbert spaces," Journal of Global Optimization, Springer, vol. 83(2), pages 249-271, June.
    5. Hongwei Liu & Jun Yang, 2020. "Weak convergence of iterative methods for solving quasimonotone variational inequalities," Computational Optimization and Applications, Springer, vol. 77(2), pages 491-508, November.
    6. Papa Quiroz, E.A. & Mallma Ramirez, L. & Oliveira, P.R., 2015. "An inexact proximal method for quasiconvex minimization," European Journal of Operational Research, Elsevier, vol. 246(3), pages 721-729.
    7. S.-M. Grad & F. Lara & R. T. Marcavillaca, 2023. "Relaxed-inertial proximal point type algorithms for quasiconvex minimization," Journal of Global Optimization, Springer, vol. 85(3), pages 615-635, March.
    8. H. Apolinário & E. Papa Quiroz & P. Oliveira, 2016. "A scalarization proximal point method for quasiconvex multiobjective minimization," Journal of Global Optimization, Springer, vol. 64(1), pages 79-96, January.

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