Computation of fixed points of symmetric extremal mappings (Q1282475)

The problem considered is to find any fixed point satisfying the inequality \[ \Phi (v^*,v^*) \leq \Phi (v^*,w),\quad \forall w \in \Omega \subset \mathbb{R}^n, \] where \(\Omega\) is a closed convex set, the function \(\Phi (v,u)\) is defined on the product \(\Omega \times \Omega\) and is symmetric, i.e. \[ \Phi (w,v) = \Phi (v,w), \quad \forall w \in \Omega, \quad \forall v \in \Omega. \] Let \(\pi_{\Omega}(v)\) be the projection of a vector \(v\) to the set \({\Omega}.\) The method of the gradient projection \[ \overline u^n= \pi_{\Omega} (v^n- \alpha \nabla \Phi_w(v^n, v^n)), \quad v^{n+1}=\pi_{\Omega} (v^n- \alpha \nabla \Phi_w(\overline u^n,\overline u^n)) \] is considered for determining the solution of the problem. Properties of symmetric functions are investigated. Existence of fixed points, convergence of this gradient projection method to the set of stationary solutions, monotone convergence with respect to the norm to one of the fixed points, convergence with the rate of a geometric progression, convergence in a finite number of steps are substantiated under suitable conditions.