Taiwanese Journal of Mathematics


Marco Castellani and Massimo Pappalardo

Full-text: Open access


We consider equilibrium problems (EP) with directionally differentiable (not necessarily ${\mathcal C}^1$) bifunctions which are convex with respect to the second variable and we use a gap function approach to solve them. In the first part of the paper we establish a condition under which any stationary point of the gap function solves (EP) and we propose a solution method which uses descent directions of the gap function. In the final section we study the problem when this condition is not satisfied. In this case we use a family of gap functions depending on a parameter $\alpha$ which allows us to overcome the trouble due to the lack of a descent direction.

Article information

Taiwanese J. Math., Volume 13, Number 6A (2009), 1837-1846.

First available in Project Euclid: 18 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 90C30: Nonlinear programming 49M37: Methods of nonlinear programming type [See also 90C30, 65Kxx]

equilibrium problems gap functions


Castellani, Marco; Pappalardo, Massimo. GAP FUNCTIONS FOR NONSMOOTH EQUILIBRIUM PROBLEMS. Taiwanese J. Math. 13 (2009), no. 6A, 1837--1846. doi:10.11650/twjm/1500405616. https://projecteuclid.org/euclid.twjm/1500405616

Export citation


  • G. Bigi, M. Castellani and M. Pappalardo, A new solution method for equilibrium problems, Optimization Methods and Software, to appear. DOI: 10.1080/10556780902855620.
  • E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63 (1993), 1-23.
  • O. Chadli, I. V. Konnov and J. C. Yao, Descent methods for equilibrium problems in a Banach space, Computers and Mathematics with Applications, 48 (2004), 609-616.
  • J. M. Danskin, The theory of min-max with applications, SIAM Journal on Applied Mathematics, 14 (1966), 641-664.
  • A. Iusem and W. Sosa, New existence results for equilibrium problems, Nonlinear Analisys, 52 (2003), 621-635.
  • I. V. Konnov and M. S. S. Ali, Descent methods for monotone equilibrium problems in Banach spaces, Journal of Computational and Applied Mathematics, 188 (2006), 165-179.
  • G. Mastroeni, On auxiliary principle for equilibrium problems, in: Equilibrium problems and variational models, P. Daniele, F. Giannessi and A. Maugeri (eds.), Kluwer Academic Publishers, Dordrecht, 2003, pp. 289-298.
  • G. Mastroeni, Gap functions for equilibrium problems, Journal of Global Optimization, 27 (2003), 411-426.