[Paper Review] A New Iterative Projection Method for Approximating Fixed Point Problems and Variational Inequality Problems

İbrahim Karahan, Murat Özdemir|arXiv (Cornell University)|Mar 13, 2014

Optimization and Variational Analysis8 references3 citations

TL;DR

This paper proposes a new extragradient iterative method for solving the common fixed point and variational inequality problem in real Hilbert spaces. By combining an infinite family of nonexpansive mappings with an inverse strongly monotone operator, the method achieves strong convergence to a common solution under mild conditions, extending existing iterative schemes with improved convergence guarantees.

ABSTRACT

In this paper, we introduce and study a new extragradient iterative process for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a variational inequality for an inverse strongly monotone mapping in a real Hilbert space. Also, we prove that under quite mild conditions the iterative sequence defined by our new extragradient method converges strongly to a solution of the fixed point problem for an infinite family of nonexpansive mappings and the classical variational inequality problem. In addition, utilizing this result, we provide some applications of the considered problem not just giving a pure extension of existing mathematical problems.

Motivation & Objective

• To develop a new iterative algorithm that simultaneously solves fixed point problems for an infinite family of nonexpansive mappings and variational inequality problems.
• To establish strong convergence of the proposed iterative sequence to a common solution in real Hilbert spaces.
• To extend existing extragradient-type methods by incorporating a hybrid structure involving two projections per iteration.
• To provide applications to convex minimization and fixed point problems involving strictly pseudocontractive mappings.

Proposed method

• The method uses a two-step iterative scheme: first, compute an intermediate point via a convex combination of the current iterate and its projection under a nonexpansive mapping; second, update the iterate using a projection of this intermediate point.
• The algorithm involves the metric projection $ P_C $, an inverse strongly monotone operator $ A $, and a nonexpansive mapping $ T $, with parameters $ \lambda_n \in [a,b] \subset (0,2\alpha) $ and $ \alpha_n \in [c,d] \subset (0,1) $.
• The key update formula is $ x_{n+1} = T P_C(I - \lambda_n A) y_n $, where $ y_n = (1 - \alpha_n)x_n + \alpha_n T P_C(I - \lambda_n A)x_n $.
• The method is designed to ensure strong convergence by leveraging the properties of nonexpansive mappings and inverse strongly monotone operators in Hilbert space.
• The convergence analysis relies on the equivalence between variational inequality solutions and fixed points of the mapping $ P_C(I - \lambda A) $.
• The method is applied to convex minimization by setting $ A = \nabla f $, where $ \nabla f $ is $ L $-Lipschitz continuous, and to fixed point problems involving strictly pseudocontractive mappings.

Experimental results

Research questions

• RQ1 Can a new iterative method be designed to simultaneously solve fixed point problems for an infinite family of nonexpansive mappings and variational inequality problems?
• RQ2 Does the proposed extragradient-type scheme achieve strong convergence under mild parameter constraints?
• RQ3 How does the convergence behavior of this method compare to existing iterative schemes like Mann, Picard, or Ishikawa iterations?
• RQ4 Can the method be extended to convex minimization problems via gradient-based formulations?
• RQ5 What is the convergence behavior when the variational inequality operator is derived from a strictly pseudocontractive mapping?

Key findings

• The proposed iterative sequence converges strongly to a point $ z \in F(T) \cap \Omega $, where $ F(T) $ is the set of fixed points of the nonexpansive mapping and $ \Omega $ is the solution set of the variational inequality problem.
• Strong convergence is guaranteed under the conditions that $ \lambda_n \in [a,b] \subset (0,2\alpha) $ and $ \alpha_n \in [c,d] \subset (0,1) $, with $ \alpha $ being the inverse strong monotonicity parameter of $ A $.
• The method achieves strong convergence even when the solution set involves an infinite family of nonexpansive mappings, which is a significant improvement over weak convergence results in prior works.
• For convex minimization, setting $ A = \nabla f $ with $ \nabla f $ being $ L $-Lipschitz continuous leads to strong convergence to the minimizer in $ \text{Argmin}_{x \in C} f(x) $, provided the solution set is nonempty.
• When applied to a pair of nonexpansive and strictly pseudocontractive mappings, the method converges weakly to a common fixed point, with convergence established via the equivalence $ F(S) = VI(C, I-S) $.
• The method is shown to be independent of and faster than classical Picard, Mann, and Ishikawa iterations in the context of contractions, as demonstrated in related work cited in the paper.