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

Abstract

We study variational inequalities governed by a point-to-set maximal monotone operator in a real Hilbert space and constrained by a convex inequality \(C=\{x\in\Hi:c(x)\le0\}\), 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 \(P_{S^*}(x^0)\), the projection of the initial point onto the solution set. We also derive best-iterate \(O(N^{-1/2})\) 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 \(\ell_1\)-subdifferential example.