Monotone Equilibrium on Hadamard Manifolds

• The paper extends equilibrium theory to Hadamard manifolds, leveraging geodesic convexity and nonpositive curvature for robust problem formulation.
• It presents intrinsic proximal-point and extragradient algorithms that adapt classical methods to nonlinear geometric settings with controlled errors.
• Convergence analyses utilize Fejér monotonicity and three-point inequalities, with applications to Nash equilibria and non-Euclidean optimization.

A monotone equilibrium problem on a Hadamard manifold is the task of finding a point in a geodesically convex subset of a manifold of nonpositive sectional curvature such that it satisfies a given monotone bifunction’s equilibrium criterion. The intrinsic geometry of Hadamard manifolds—unique geodesics, global nonpositive curvature, and global convexity of the squared distance—enables the extension of monotone operator and equilibrium-problem methodologies from linear spaces to a nonlinear geometric context. This article provides a rigorous account of the definition, structure, monotonicity notions, algorithmic frameworks, convergence theory, and major advances in the study of monotone equilibrium problems on Hadamard manifolds.

1. Geometry of Hadamard Manifolds and Problem Formulation

Let $M$ be a finite-dimensional Hadamard manifold: a complete, simply connected Riemannian manifold with everywhere nonpositive sectional curvature. For any $x,y \in M$, there is a unique minimal geodesic $\gamma$ from $x$ to $y$; the exponential map $\exp_x:T_x M \to M$ and its inverse $\exp_x^{-1}:M \to T_x M$ are global diffeomorphisms, and the Riemannian distance $d(x,y)$ is defined via geodesic length.

A nonempty, closed subset $C \subset M$ is geodesically convex if every geodesic connecting two points in $C$ lies entirely in $C$. A real-valued function is geodesically convex if its restriction to any geodesic is convex in the usual sense.

Given $C \subset M$ and bifunction $F:C\times C \to \mathbb{R}$, the equilibrium problem (EP) is to find $x^* \in C$ such that

$F(x^*, y) \geq 0 \quad \forall y \in C.$

The canonical examples include variational inequalities for monotone vector fields, Nash equilibrium models on curved strategy sets, and convex-constrained minimization problems (Fan et al., 2020, Kumam et al., 2018, Khatibzadeh et al., 2016, Kristály, 2016).

2. Notions of Monotonicity and Regularity

Monotonicity generalizes classical operator monotonicity to the manifold context. The standard definitions for a bifunction $F:C\times C\to \mathbb{R}$ are as follows (Fan et al., 2020, Sharmaa et al., 27 Jun 2025):

• Monotone: $F(x,y) + F(y,x) \leq 0$ for all $x,y \in C$.
• Strongly monotone: There exists $\gamma>0$ such that $F(x,y) + F(y,x) \leq -\gamma d^2(x,y)$.
• Pseudomonotone: $F(x,y)\geq 0 \implies F(y,x) \leq 0$ for all $x,y \in C$.
• Strongly pseudomonotone: There exists $\gamma>0$ such that $F(x,y)\geq 0 \implies F(y,x)\leq -\gamma d^2(x,y)$.

These monotonicity properties are crucial for establishing well-posedness, uniqueness, and convergence of algorithms. Notably, strong monotonicity implies monotonicity, which in turn implies pseudomonotonicity (Fan et al., 2020).

A weaker but widely used smoothness assumption is a Lipschitz-type growth condition in $F$:

$$F(x,y)+F(y,z) \geq F(x,z) - \gamma_1 d^2(x,y) - \gamma_2 d^2(y,z), \quad \forall x,y,z \in C.$$

Coercivity-type conditions are often needed to rule out minimization at infinity in noncompact domains (Khatibzadeh et al., 2016, Bento et al., 2021).

3. Algorithmic Frameworks: Proximal and Extragradient Methods

Algorithmic methods for monotone equilibrium problems on Hadamard manifolds generalize classical extragradient and proximal-point methods to the non-Euclidean setting (Fan et al., 2020, Sharmaa et al., 27 Jun 2025, Kumam et al., 2018, Sharma et al., 20 Jan 2026, Batista et al., 2015, Bento et al., 2021). The metric and curvature structure dictate the form of the regularizers and update rules:

a. Proximal-Point Algorithm (PPA): For monotone $F$, define the resolvent $J_\lambda$ by

$$J_\lambda(x) = \operatorname*{argmin}_{y\in C} \left\{ \lambda F(x,y) + \frac{1}{2} d^2(x, y) \right\},$$

and iterate $x_{n+1} = J_{\lambda_n}(x_n)$ (Kumam et al., 2018, Khatibzadeh et al., 2016). This step is intrinsic to the manifold and leverages global convexity of the squared-distance.

b. Extragradient-type Methods: These algorithms introduce auxiliary points:

$$\begin{align*} y_n &= \operatorname*{argmin}_{y\in C} \{ F(x_n, y) + (2\lambda_n)^{-1} d^2(x_n, y) \}, \ x_{n+1} &= \operatorname*{argmin}_{y\in C} \{ F(y_n, y) + (2\lambda_n)^{-1} d^2(x_n, y) \}, \end{align*}$$

with suitable variable stepsize selection (Fan et al., 2020). Regularized extragradient methods use a Busemann function or squared-distance difference for regularization, yielding algorithms that are robust even without Lipschitz continuity of $F$ (Sharmaa et al., 27 Jun 2025, Bento et al., 2021).

c. Inexact and Enlarged Methods: Proximal and extragradient iterates may be computed inexactly via enlargement of maximal monotone vector fields and controlled errors, yielding convergence under summable error tolerances (Bento et al., 2021, Batista et al., 2015, Batista et al., 2018).

d. Bregman-Regularized PPA: For some applications, the regularization is performed using a Bregman divergence $D_\phi(x, y)$ arising from a geodesically convex function $\phi$, leading to schemes of the form

$$x_{n+1} = \operatorname*{argmin}_{x\in C} \left\{ \lambda_n F(x, x_n) + D_\phi(x, x_n) \right\},$$

along with an appropriately regularized bifunction (Sharma et al., 20 Jan 2026).

4. Convergence Properties

The nonpositive curvature of Hadamard manifolds is essential for global convergence analyses. The central arguments rely on Fejér monotonicity, quasi-Fejér monotonicity, and generalized three-point inequalities, allowing direct generalization of Euclidean results.

Global Convergence: Under monotonicity, geodesic convexity, suitable continuity, and coercivity conditions, both PPA and extragradient sequences converge to solutions of EP$(F,C)$. Key properties include:

• Fejér monotonicity relative to the solution set (Sharmaa et al., 27 Jun 2025, Fan et al., 2020, Kumam et al., 2018, Batista et al., 2015).
• Asymptotic regularity: $d(x_{n+1}, x_n)\to 0$.
• Cluster point analysis is refined using metric geometry arguments unique to Hadamard spaces (Khatibzadeh et al., 2016).

Linear Rate under Strong Monotonicity: For strongly pseudomonotone or strongly monotone problems, one obtains $R$-linear convergence rates by descent estimates tailored to the manifold geometry:

$d^2(x_{n+1}, x^*) \leq (1 - \rho) d^2(x_n, x^*),$

for some $\rho\in (0,1)$, yielding geometric decay of errors (Fan et al., 2020, Sharmaa et al., 27 Jun 2025).

Global Error Bounds: For strongly pseudomonotone bifunctions, global error bounds relate residuals $d(x_n, x^*)$ to iterative errors $d(x_n, y_n)$, facilitating stopping criteria and complexity analysis (Sharmaa et al., 27 Jun 2025).

Limitations: Sublinear or $\Delta$-convergence can occur for non-strongly monotone cases; additional regularizations (e.g., Halpern steps) may be used to enforce strong convergence (Khatibzadeh et al., 2016).

5. Variational Inequality Formulation and Maximal Monotone Fields

For monotone equilibrium problems, the variational inequality (VI)

$$\text{Find } x^* \in C : \exists u^* \in X(x^*) \text{ with } \langle u^*, \exp_{x^*}^{-1} y \rangle \geq 0, \forall y \in C,$$

where $X$ is a maximal monotone vector field, is equivalent to the EP for the bifunction $F$. This equivalence enables the transference of operator-theoretic algorithms (proximal-point, enlargement) to equilibrium problems (Bento et al., 2021, Batista et al., 2018, Batista et al., 2015).

On Hadamard manifolds:

• The exponential map and parallel transport are used for constructing monotonicity and normal cone mappings intrinsically.
• The CAT(0) geometry ensures projection operators and feasibility arguments hold with the same force as in Hilbert space (Kristály, 2016, Kumam et al., 2018).

6. Notable Algorithms and Implementation Considerations

The most current developments include extragradient methods whose regularization is performed either via Busemann function terms (Bento et al., 2021) or squared-distance regularizations (Sharmaa et al., 27 Jun 2025), with well-posedness ensured by the geodesic convexity properties unique to Hadamard manifolds.

A summary of principal algorithmic schemes is presented below:

Algorithm Class	Update Structure	Notable Features
Proximal-Point	$x_{n+1} = J_\lambda(x_n)$	Firm nonexpansiveness, single-valued steps
Extragradient	$x_{n+1}$, $y_n$ via successive regularized subproblems	No linesearch; variable stepsizes
Inexact/Enlarged PPA	Enlargement $X^{\varepsilon}$, errors $e_n$	Robust to errors; summable error tolerances
Bregman PPA	$$x_{n+1} = \arg\min\{ \lambda_n F(\cdot, x_n) + D_\phi \}$$	Handles non-Euclidean divergence penalties

Practical implementation exploits the closed-form expressions for exponential/logarithmic maps and distances in symmetric spaces (e.g., $\mathbb{R}^n_{++}$ with Log-Euclidean metrics) and can leverage manifold-optimization toolboxes for convex-constrained subsolvers (Fan et al., 2020, Sharma et al., 20 Jan 2026, Sharmaa et al., 27 Jun 2025).

7. Applications and Outlook

Monotone equilibrium problems on Hadamard manifolds arise in Nash and generalized Nash equilibrium on curved strategy sets, non-Euclidean traffic flow and network equilibrium, geometric imaging and machine learning models with geodesic constraints, and matrix-valued optimization (e.g., symmetric positive definite matrices with trace-type metrics) (Kristály, 2016).

Recent research demonstrates the utility of these frameworks for high-dimensional statistical manifolds, minimax formulations, and nonconvex Bregman settings, with robust convergence properties and practical performance even for large-scale structured problems (Sharmaa et al., 27 Jun 2025, Sharma et al., 20 Jan 2026).

Ongoing directions include extension to non-Hadamard (positive or variable curvature) settings, refinement of complexity bounds, adaptive and inertial variants, and further exploration of Bregman and nonconvex regularization (Batista et al., 2018).

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References:

• "An explicit extragradient algorithm for equilibrium problems on Hadamard manifolds" (Fan et al., 2020)
• "Regularized Extragradient Methods for Solving Equilibrium Problems on Hadamard Manifolds" (Sharmaa et al., 27 Jun 2025)
• "A Bregman Regularized Proximal Point Method for Solving Equilibrium Problems on Hadamard Manifolds" (Sharma et al., 20 Jan 2026)
• "Equilibrium Problems and Proximal Algorithms in Hadamard Spaces" (Kumam et al., 2018)
• "Enlargement of Monotone Vector Fields and an Inexact Proximal Point Method for Variational Inequalities in Hadamard Manifolds" (Batista et al., 2015)
• "An inexact proximal point method for variational inequality on Hadamard manifolds" (Bento et al., 2021)
• "Elements of Convex Geometry in Hadamard Manifolds with Application to Equilibrium Problems" (Bento et al., 2021)
• "An Extragradient-type Algorithm for Variational Inequality on Hadamard Manifolds" (Batista et al., 2018)
• "Monotone and Pseudo-Monotone Equilibrium Problems in Hadamard Spaces" (Khatibzadeh et al., 2016)
• "Splitting Algorithms of Common Solutions Between Equilibrium and Inclusion Problems on Hadamard Manifolds" (Khammahawong et al., 2019)
• "Nash-type equilibria on Riemannian manifolds: a variational approach" (Kristály, 2016)