Two accelerated double-direction methods for convex-constrained nonlinear equations with applications

This paper presents two novel double-direction methods for addressing large-scale nonlinear equations with convex constraints. The first approach employs the Frobenius norm to calculate the discrepancy between Broyden’s method and its approximation, allowing us to derive an acceleration parameter. The second approach introduces a correction parameter through a hybrid iterative procedure that integrates the Picard-Mann methods in its search direction. The algorithm meets the sufficient descent condition without requiring any line search. We established both global convergence and R-linear convergence of the method under favorable conditions. Numerical simulations demonstrate the effectiveness of the proposed algorithms. Furthermore, the method has been successfully applied to restore blurred images, highlighting its practical relevance.