Gradient approach of computing fixed points of equilibrium problems
The author considers the general equilibrium problem: Find \(v^*\in\Omega_0\) such that \[ \Phi(v^*,w)\geq\Phi (v^*, v^*) \quad \forall w \in \Omega _{0}, \] where \(\Phi : \mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R}\) is a differentiable bifunction which is convex in the second variable and \( \Omega _{0}\) is a convex closed subset of \(\mathbb{R}^n\). Then it is equivalent to the variational inequality which is to find \(v^* \in \Omega _{0}\) such that \[ \langle F (v^*), w-v^* \rangle \geq 0 \quad \forall w \in \Omega _{0}, \] where \(F(v) = \nabla _{w} \left.\Phi (v,w) \right | _{w=v}\). The author considers several conditions on \(\Phi \) and the Jacobian of \(F\) which provide the integrability of \(F\) and the convergence of the extragradient method. Its convergence is also proved under monotonicity type assumptions on \(\Phi \). Some rates of convergence, which are similar to the previous ones, see e.g. \textit{A. S. Antipin} [Comput. Math. Math. Phys. 35, 539--551 (1995; Zbl 0852.65046)], are also given.
scientific article; zbMATH DE number 1924945