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Proximal and dynamical approaches to equilibrium problems. (English) Zbl 0944.65080

Théra, Michel (ed.) et al., Ill-posed variational problems and regularization techniques. Proceedings of a workshop, Univ. of Trier, Germany, September 3-5, 1998. Berlin: Springer. Lect. Notes Econ. Math. Syst. 477, 187-201 (1999).
From the introduction: The theory of equilibrium problems has emerged as an interesting branch of applied mathematics, permitting the general and unified study of a large number of problems arising in mathematical economics, optimization and operations research. Inspired by numerical methods developed for variational inequalities and motivated by recent advances in this field, we propose several ways (including an auxiliary problem principle, a selection method, as well as a dynamical procedure) to solve the following equilibrium problem: \[ \text{Find}\quad \overline x\in C\quad\text{such that}\quad F(\overline x,x)+ \langle G(\overline x),x-\overline x\rangle\geq 0\quad\forall x\in C, \] where \(C\) is a nonempty convex closed subset of a real Hilbert space \(X\), \(F:C\times C\to\mathbb{R}\) is a given bivariate function with \(F(x,x)= 0\) for all \(x\in C\) and \(G: C\to\mathbb{R}\) is a continuous mapping. This problem has useful applications in nonlinear analysis, including as special cases optimization problems, variational inequalities, fixed point problems and problems of Nash equilibria.
For the entire collection see [Zbl 0930.00059].

MSC:

65K10 Numerical optimization and variational techniques
49J40 Variational inequalities
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