An Interior Proximal Method for a Class of Quasimonotone Variational Inequalities
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The Bregman-function-based Proximal Point Algorithm for variational inequalities is studied. Classical papers on this method deal with the assumption that the operator of the variational inequality is monotone. Motivated by the fact that this assumption can be considered to be restrictive, e.g., in the discussion of Nash equilibrium problems, the main objective of the present paper is to provide a convergence analysis only using a weaker assumption called quasimonotonicity. To the best of our knowledge, this is the first algorithm established for this general and frequently studied class of problems.
KeywordsQuasimonotone operators Variational inequalities Bregman distances Proximal Point Algorithm Interior-point-effect
I am grateful to an anonymous referee whose comments greatly improved the present paper. Moreover, my thanks also go to the associate editor and the editor-in-chief for further very helpful remarks.
- 12.Polyak, B.T.: Introduction to Optimization. Optimization Software, New York (1987) Google Scholar
- 13.Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) Google Scholar
- 16.Rockafellar, R., Wets, R.: Variational Analysis. Springer, Berlin (1997) Google Scholar
- 28.Langenberg, N.: Convergence Analysis of Proximal-Like Methods for Variational Inequalities and Fixed Point Problems Applications to the Nash Equilibrium Problem. Logos, Berlin (2011) Google Scholar
- 32.Castellani, M., Giuli, M.: A characterization of the solution set of pseudoconvex extremum problems. J. Convex Anal. 19 (2013) Google Scholar