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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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).