---
title: Monotone Equilibrium on Hadamard Manifolds
url: https://www.emergentmind.com/topics/monotone-equilibrium-problems-on-hadamard-manifolds
type: topic
---

# Monotone Equilibrium on Hadamard Manifolds

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 [2005.10444, 1807.10900, 1611.01829, 1602.04157]. 

## 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 [2005.10444, 2506.22391]:

- **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 [2005.10444].

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 [1611.01829, 2107.02223].

## 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 [2005.10444, 2506.22391, 1807.10900, 2601.13959, 1511.08301, 2103.02116]. 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 $x_{n+1} = J_{\lambda_n}(x_n)$ [1807.10900, 1611.01829]. 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 [2005.10444]. Regularized extragradient methods use a Busemann function or squared-distance difference for regularization, yielding algorithms that are robust even without Lipschitz continuity of $F$ [2506.22391, 2107.02223].

**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 [2103.02116, 1511.08301, 1804.09292].

**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 [2601.13959].

## 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 [2506.22391, 2005.10444, 1807.10900, 1511.08301].
- Asymptotic regularity: $d(x_{n+1}, x_n)\to 0$.
- Cluster point analysis is refined using metric geometry arguments unique to Hadamard spaces [1611.01829].

**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 [2005.10444, 2506.22391].

**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 [2506.22391].

**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 [1611.01829].

## 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 [2103.02116, 1804.09292, 1511.08301].

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 [1602.04157, 1807.10900].

## 6. Notable Algorithms and Implementation Considerations

The most current developments include extragradient methods whose regularization is performed either via Busemann function terms [2107.02223] or squared-distance regularizations [2506.22391], 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 [2005.10444, 2601.13959, 2506.22391].

## 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) [1602.04157].

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 [2506.22391, 2601.13959].

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 [1804.09292].

---

**References:**
- "An explicit extragradient algorithm for equilibrium problems on Hadamard manifolds" [2005.10444]
- "Regularized Extragradient Methods for Solving Equilibrium Problems on Hadamard Manifolds" [2506.22391]
- "A Bregman Regularized Proximal Point Method for Solving Equilibrium Problems on Hadamard Manifolds" [2601.13959]
- "Equilibrium Problems and Proximal Algorithms in Hadamard Spaces" [1807.10900]
- "Enlargement of Monotone Vector Fields and an Inexact Proximal Point Method for Variational Inequalities in Hadamard Manifolds" [1511.08301]
- "An inexact proximal point method for variational inequality on Hadamard manifolds" [2103.02116]
- "Elements of Convex Geometry in Hadamard Manifolds with Application to Equilibrium Problems" [2107.02223]
- "An Extragradient-type Algorithm for Variational Inequality on Hadamard Manifolds" [1804.09292]
- "Monotone and Pseudo-Monotone Equilibrium Problems in Hadamard Spaces" [1611.01829]
- "Splitting Algorithms of Common Solutions Between Equilibrium and Inclusion Problems on Hadamard Manifolds" [1907.00364]
- "Nash-type equilibria on Riemannian manifolds: a variational approach" [1602.04157]

Source: https://www.emergentmind.com/topics/monotone-equilibrium-problems-on-hadamard-manifolds