A Hybrid Fixed Point Algorithm for Solving Convex Optimization Problems

Abstract

This paper introduces a novel hybrid fixed point iterative algorithm aimed at solving convex optimization problems in real Hilbert spaces. The algorithm is designed to find a solution that lies in the intersection of a closed convex set and the fixed point set of a nonexpansive mapping, a common formulation in various applied mathematical and engineering contexts. By integrating projection operators with averaged nonexpansive mappings and gradient-based updates, the proposed scheme ensures strong convergence to an optimal solution under standard and practically verifiable assumptions on the step size parameters. Unlike traditional projection methods or purely fixed point iterations, our hybrid approach leverages the geometric structure of Hilbert spaces to guarantee convergence even in the absence of strong monotonicity or Lipschitz continuity. The theoretical development is supported by rigorous convergence analysis, which confirms that the generated sequence converges strongly to a point that minimizes a given convex objective function over the fixed point set of a nonexpansive mapping. A detailed numerical example in $\mathbb{R}^2$ is provided to illustrate the algorithm’s practical behavior and convergence characteristics, demonstrating its stability and efficiency. The results not only validate the theoretical claims but also highlight the flexibility of the algorithm for potential applications in areas such as signal processing, machine learning, and variational inequalities. Furthermore, our framework unifies and generalizes several well-known iterative schemes in the literature, thereby contributing a fresh perspective and an effective computational tool for solving constrained convex optimization problems through fixed point techniques.