An Inertial Self-Adaptive Tseng Method for Non-Lipschitz Monotone Variational Inclusions on Hadamard Manifolds

We study the monotone variational inclusion 0∈Φ(x)+ψ(x) on a finite-dimensional Hadamard manifold M, where Φ is a single-valued monotone vector field and ψ is maximal monotone. Two features distinguish the present work from the existing Riemannian Tseng-type literature. First, Φ is assumed only uniformly continuous on bounded sets, not Lipschitz. Second, and more substantially, we show that the parallel-transport form of Tseng’s correction step, which is the form used throughout that literature, does not satisfy the inequality from which the descent estimate is derived there: the Euclidean identity it relies on has no curved counterpart. We give an explicit configuration in hyperbolic space where that inequality fails and the resulting descent inequality fails by more than 1% of d(w,x*)2. We then propose a corrected correction step, built entirely inside a single tangent space and using no parallel transport, which reduces to Tseng’s step in the Euclidean case and for which the cross terms cancel exactly as they do in Hilbert space. Assuming a two-sided curvature bound −κ2≤K≤0, we prove a descent inequality whose only curvature cost is the comparison factor ζ(κ,t)=κt/tanh(κt) multiplying the squared length of the correction, and we show this constant is sharp. The resulting step-size condition is τ<ζ(κ,D)−1/2, which recovers the Hilbert-space condition τ<1 as κ→0. Strong convergence to the point of Ω singled out, in the viscosity sense, by a contraction h:M→M is established, under an a priori bound D on the resolvent iterates and non-degeneracy of the self-adaptive step sizes. Numerical experiments on the manifold of symmetric positive definite matrices and on hyperbolic space accompany the theory.