A Projection-Free Algorithm for Variational Inequalities in Hilbert Spaces with Strong Convergence

We study variational inequalities governed by a point-to-set maximal monotone operator in a real Hilbert space and constrained by a convex inequality $C=\{x\in \text{Hi}:c(x)\le 0\}$, where the defining function $c$ is continuous and not necessarily differentiable. The proposed method uses only projections onto intersections of half-spaces and avoids the metric projection onto $C$. Feasibility is handled by subgradient cuts and, when a trial operator point is infeasible, by a Slater correction based on a fixed strictly feasible point. The variational inequality is represented by Minty-type separating half-spaces generated at feasible graph points of the operator, and a Haugazeau half-space is added to obtain best-approximation convergence. Under a Slater-corrected feasible-separation condition, together with explicit exact, approximate and finite-candidate oracle realisations, the whole sequence converges strongly to $P_{S^{\ast }}(x^0)$, the projection of the initial point onto the solution set. We also derive best-iterate $O(N^{-1\mathrm{/}2})$ residual estimates for the step residual, feasibility violation and Minty gap. The analysis is stated directly for point-to-set maximal monotone operators, while the concrete oracle realisations include finite-dimensional single-valued models. We record the consequences of strong monotonicity in the point-to-set setting and provide numerical comparisons on nonsmooth and large-scale constraints, including maxima of convex quadratics, a discretised optimal-control problem, mixed-norm sparse recovery, a Cournot–Nash capacity equilibrium, and a genuine point-to-set ${\mathrm{\ell }}_1$-subdifferential example.