Iterative Methods for Variational Inequalities in Hilbert Spaces

Variational inequalities in Hilbert spaces form a unifying framework for many equilibrium, optimisation and complementarity problems. Given a closed convex set and a monotone operator on a real Hilbert space, the aim is to find a point in the set whose inner product with any feasible direction under the operator is non-negative. Iterative methods for these problems exploit the geometry of Hilbert spaces—most notably the existence of orthogonal projections—to generate sequences that converge to a solution. Classical approaches include the projection method, which uses simple metric projections, and the extragradient method, which introduces an auxiliary step to handle non-Lipschitz perturbations and pseudo-monotonicity. More recent advances weave in inertial or momentum terms to accelerate convergence, relaxation parameters to ensure stability under weaker assumptions, and splitting techniques—such as forward–backward and proximal–point schemes—to decompose complex operators into simpler subproblems. Hybrid and block-iterative frameworks further allow parallel or coordinate-wise updates, enhancing scalability. Across applications in image processing, network equilibrium, signal reconstruction and machine learning, these algorithms have demonstrated robustness in handling large-scale and noisy data, while theoretical analysis has established both weak and strong convergence under broad operator conditions.

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Recent studies have refined the extragradient method to cover pseudo-monotone variational inequalities in infinite-dimensional Hilbert spaces. By carefully selecting step sizes and employing a two-step projection correction, the resulting algorithm is shown to generate a sequence that converges weakly to a solution under only pseudo-monotonicity and Lipschitz continuity. Numerical experiments on variational models arising in contact mechanics and economics demonstrate practical efficiency compared with classical projection schemes.

Advances in split feasibility problems—where one seeks a point in one Hilbert space whose image under a bounded linear operator lies in another convex set—have been driven by inertial and relaxation techniques. A modified inertial relaxed CQ algorithm introduces an alternating inertial step that combines the Krasnosel’skiĭ–Mann method with the classic CQ approach. Under mild assumptions on the operator norm and relaxation parameters, strong convergence to a solution is proved. Applications to sparse signal recovery and image deblurring illustrate accelerated convergence and reduced computational effort in high-dimensional settings.

Building on inertial ideas, a relaxed CQ scheme with alternated inertial steps has been proposed for general split feasibility problems. By incorporating an additional inertial momentum term and adaptive relaxation, the algorithm achieves improved convergence speed without sacrificing stability. Numerical tests in medical imaging and wireless network design confirm that the inertial version outperforms its non-inertial predecessor, particularly in ill-conditioned scenarios.

Iterative Methods for Variational Inequalities in Hilbert Spaces publication trend

The graph below shows the total number of articles in iterative methods for variational inequalities in hilbert spaces across all publications each year (not limited to Nature Index journals).

Technical terms

Hilbert space: A complete inner-product space in which notions of angle, orthogonality and projection are defined.

Variational inequality: A problem of finding x in a convex set C such that ⟨F(x), y–x⟩ ≥ 0 for all y in C, where F is an operator on the space.

Monotone operator: An operator F for which ⟨F(x)–F(y), x–y⟩ ≥ 0 holds for all x, y, ensuring stability of solution methods.

Extragradient method: A two-step projection algorithm that first computes an auxiliary iterate to handle non-Lipschitz or pseudo-monotone behaviours, then corrects via a second projection.

Relaxation parameter: A scalar weight applied to update steps to ensure convergence under weaker assumptions on operator Lipschitz constants.

Inertial (momentum) method: An approach that adds a weighted difference of successive iterates to accelerate convergence, borrowing ideas from heavy-ball and Nesterov schemes.

Split feasibility problem: A problem of finding x in C such that Ax belongs to another convex set D, where A is a bounded linear operator between Hilbert spaces.

References

1. On the Weak Convergence of the Extragradient Method for Solving Pseudo-Monotone Variational Inequalities. Journal of Optimization Theory and Applications (2018).
2. The modified inertial relaxed CQ algorithm for solving the split feasibility problems. Journal of Industrial and Management Optimization (2018).
3. New inertial relaxed method for solving split feasibilities. Optimization Letters (2020).