1 Introduction↩︎

The subgradient method is one of the classical algorithms for a first-order solution of nondifferentiable convex optimization problems. For a convex function \(f:\mathbb{R}^n \to \mathbb{R}\), one may select at iteration \(k\) a subgradient \(g^k \in \partial f(x^k)\) and perform the update \[\label{eq:subgradient} x^{k+1} = x^k - \alpha_k g^k, \qquad k=0,1,2,\ldots,\tag{1}\] with a suitable step-size sequence \(\{\alpha_k\}\). This classical Euclidean form was proposed by Shor in the 1960s and has been extensively studied; see, e.g., [1]–[7] and references therein. The subgradient method is also a simple way to address convex feasibility problems, such as the classical Convex Feasibility Problem (CFP); see, for instance, [8]–[10].

In the Riemannian setting, the CFP consists of finding a point in the intersection of finitely many geodesically convex sets, which are often given as sublevel sets of convex functions. In the manifold context, this problem is further motivated by the fact that certain feasibility problems that are nonconvex in the usual Euclidean sense may become convex after endowing the domain with a suitable Riemannian metric. To the best of our knowledge, a first systematic analysis of a subgradient-type scheme for CFPs on manifolds was carried out by Bento and Melo [11]. In the aforementioned work, a subgradient algorithm is proposed and shown to converge to a feasible point under the assumption that the manifold has nonnegative sectional curvature. The influence of curvature was already noted in [12], [13], where the authors presented a counterpart of the gradient (resp. subgradient) algorithm for optimization on manifolds, explicitly discussing the role of sectional curvature in the method’s convergence and proving convergence under the assumption of nonnegative sectional curvature. See also [14], [15], where subgradient-type schemes for convex minimization on Riemannian manifolds are explored and also focus on the case of nonnegative sectional curvature using comparison arguments to obtain Fejér-type estimates.

As far as we know, the first approaches to dealing with subgradient-type schemes on Hadamard manifolds with lower bounded curvature trace back to [16], [17], which addressed the challenge of solving the feasibility problem on Riemannian manifolds with negative sectional curvature posed in [11]. These works have served as a reference for subsequent research on first-order methods on manifolds with lower bounded curvature; see, for example, [18]–[21] and the references therein. Restricting ourselves to subgradient-type schemes for convex minimization on Riemannian manifolds, Wang [21] was among the first to study subgradient algorithms for convex optimization on complete Riemannian manifolds whose sectional curvatures are bounded below. The analysis and convergence properties are established considering step-size rules that include the classical diminishing steps: \[\sum_{k}\alpha_k=\infty\quad\text{and}\quad\sum_{k}\alpha_k^2<\infty.\] Building on this framework, Ferreira et al. [19] went further by providing iteration-complexity bounds in addition to asymptotic analysis. Despite these advances, it remains natural to ask to what extent classical Euclidean convergence mechanisms for the subgradient method can be transferred to the Riemannian setting under lower bounded curvature.

The subgradient method 1 , analyzed by Shepilov [7], employs a (non-summable diminishing) step-size condition stated in terms of \[\alpha_k \ge 0, \quad \lim_{k\to\infty}\alpha_k = 0, \quad \sum_{k=1}^{\infty}\alpha_k= \infty.\] Assuming that the set \(S\) of minimizers of \(f\) is nonempty, it was proved that \(d(x^k,S) \to 0\) provided at least one of the following additional conditions holds: \(\{x^k\}\) is bounded; \(S\) is bounded; \(S\) is a linear manifold in \(\mathbb{R}^n\); \(n=2\). When \(\sum \lambda_k^2 < \infty\) the sequence \(\{x^k\}\) converges to a point in \(S\). Furthermore, if \(S\) has a nonempty interior, then the subgradient method terminates in a finite number of steps.

Motivated by these developments and by Shepilov’s classical Euclidean analysis, we study convergence of the subgradient method on Hadamard manifolds with lower bounded curvature. Assuming the solution set \(S\) is nonempty, we establish that \(d(x^k,S) \to 0\) under the (non-summable diminishing) step-size condition expressed whenever at least one of the following holds:

1. the sequence \(\{x^k\}\) is bounded;

2. the solution set \(S\) is bounded;

Furthermore, we have the convergence of the iterates when at least one of the following holds:

1. \(\displaystyle \sum_{k=1}^{\infty}\lambda_k^2<\infty\);

2. \(\operatorname{int}(S)\neq\emptyset\), in which case the method terminates in finitely many steps.

The main contribution of this paper is a Riemannian counterpart of Shepilov’s Euclidean convergence analysis for the subgradient method on manifolds with lower bounded curvature, complementing recent advances in first-order Riemannian optimization for convex minimization and convergence analysis under lower bounded curvature.

The paper is organized as follows. In Section 2 we collect fundamental definitions and auxiliary results from Riemannian geometry that are essential for our study. The section also presents elements of convex analysis on Hadamard manifolds, including an original contribution, which is an interesting result on its own and provides a useful technical tool for subsequent convergence analysis and for future first-order methods on manifolds. In Section 3, we revisit the subgradient method and provide further results on its convergence. Finally, the last section presents the final considerations.

2 Preliminares↩︎

In this section, we present some pertinent concepts and results related to Riemannian geometry. For more details, see, for example, [22]–[25].

Let \(M\) be a complete and connected Riemannian manifold. It is known that \((M,d)\) is a complete metric space, where \(d\) denotes the Riemannian distance. We denote by \(T_xM\) the tangent space of \(M\) at \(x\in M\) and by \(TM\) the tangent bundle of \(M\) set to be those pairs \(\theta=(x,v),\) \(x\in M, \; v\in T_xM\). The Riemannian metric of \(M\) is denoted by \(\langle \cdot, \cdot \rangle\), with the corresponding norm given by \(\| \cdot \|\). For \(\theta=(x,v)\in TM\), \(\gamma_{\theta}(\cdot)\) denotes the unique geodesic with initial conditions \(\gamma_{\theta}(0)=x\) and \(\gamma_{\theta}'(0)=v\). Let us denote by \(exp:TM\to M\), \(exp(\theta):=\gamma_{\theta}(1)\), the exponential map. For each fixed \(x\in M\), \(exp_xv :=\exp(\theta)\) where \(\theta = (x,v)\). The Levi-Civita connection associated with the Riemannian manifold \((M,{\langle} \cdot,\cdot {\rangle})\) is denoted \(\nabla\). For a smooth function on \(M\), the metric induces its gradient denoted by \(grad f\). The restriction of a geodesic to a closed bounded interval is called a geodesic segment. If \(\gamma\) is a geodesic segment joining points \(x\) and \(y\) in \(M\) then, for each \(t\in [a,b]\), \(\nabla\) induces a linear isometry, relative to \({ \langle}\cdot ,\cdot {\rangle}\), \(P_{\gamma(a)\gamma(t)}:T_{\gamma(a)}M\to T_{\gamma(t)}M\), the so-called parallel transport along \(\gamma\) from \(\gamma(a)\) to \(\gamma(t)\). In the particular case where \(\gamma\) is the unique geodesic segment joining \(x\) and \(y\), then the parallel transport along \(\gamma\) from \(x\) to \(y\) is denoted by \(P_{xy}:T_{x}M\to T_{y}M\). For each \(k\) positive, let us consider the Riemannian manifold \((M, \langle \,\cdot , \cdot\, \rangle_k)\) where \(\langle u, v \rangle_k := \displaystyle\frac{1}{k^2}\langle u, v \rangle\) for any \(u,v \in T_xM\). In particular, \(||u||_k = \displaystyle\frac{1}{k}||u||\) where \(|| \cdot||_k\) denotes the norm induced by the metric \(\langle \,\cdot , \cdot\, \rangle_k\). These metrics are known as conformal metrics. We will denote the Riemannian manifold \((M, \langle \,\cdot , \cdot\, \rangle_k)\) by \(M_k\) and by \(\nabla_k\) the Levi-Civita connection associated with the Riemannian manifold \(M_k\). It is known that \(\nabla_k = \nabla\) and, consequently, the geodesics of \(M\) and \(M_k\) coincide and \(d_k = \displaystyle\frac{1}{k} d\) where \(d\) and \(d_k\) are, respectively, the distances in \(M\) and \(M_k\). Denoting by \(\exp_x^k\) the exponential map in \(M_k\), one has \(\exp_x^k v := \exp_x v\). Furthermore, the sectional curvatures of \(M\) and \(M_k\) satisfy the following equality \[K_x^k(w,\tilde{w}) = k^2K_x(w,\tilde{w}),\] where \(w,\tilde{w} \in T_xM\) are linearly independent vectors. For a unitary vector \(v \in T_x M\), note that \(\tilde{v}:=kv\) is a unitary vector with respect to the metric of \(M_k\).

It is well known that the Poincaré hyperbolic disk \(\mathbb{D}\) is a Hadamard manifold with constant sectional curvature equal to \(-1\), which will be discussed later. In \(\mathbb{D}\), the law of cosines for a triangle with side lengths \(a, b, c\) and with angle opposite to side \(a\) denoted by \(\alpha\) is given by: \[\cosh(a) = \cosh(b)\cosh(c) - \sinh(b)\sinh(c) \cos\alpha.\] Note that \(\mathbb{D}_\kappa\) is a surface with constant sectional curvature equal to \(-\kappa^2\) and, given a triangle in \(\mathbb{D}\), there exists a corresponding triangle in \(\mathbb{D}_\kappa\) whose angles are the same and the length of the sides are divided by \(\kappa\) with respect to conformal metric. In particular, given a triangle in \(\mathbb{D}\kappa\) with side lengths \(a',b',c'\) and angle opposite to \(a'\) denoted by \(\alpha\), one has the following: \[\cosh (\kappa a') = \cosh (\kappa b')\cosh (\kappa c') - \sinh (\kappa b')\sinh ( \kappa c') \cos \alpha.\] As an application of the Rauch comparison theorem (see the proof of [23]), we get the following comparison version of the law of cosines for Hadamard manifolds whose sectional curvature is bounded below and/or above.

2.1 Convex Analysis↩︎

In this section, we introduce the basic definitions, notation, and key properties of convex functions on Riemannian manifolds. We begin by presenting Busemann functions as an important class of convex examples and highlight an explicit formula for their gradient on the Poincaré disk, which both illustrates our method and suggests directions for future work. Afterwards, we recall the notion of the subdifferential for a convex function and summarize its main properties; for further details, see [11], [17], [26], [27].

A function \(f\colon M \to \mathbb{R}\) is said to be convex if and only if, for every geodesic segment \(\gamma\colon [a,b]\to M\), the composition \(f\circ \gamma\colon [a,b]\to \mathbb{R}\) is convex (in the usual sense). It is known that if \(f\colon M\to\mathbb{R}\) is convex, then \(f\) is continuous on \(M\).

An important class of convex functions on nonpositively curved spaces is provided by Busemann functions. Given \(p\in M\), \(v\in T_pM\), \(\|v\|=1\), and \(\gamma:[0,\infty)\to M\) a unit-speed geodesic starting at \(p\) with \(v=\gamma'(0)\in T_pM\), for \(x\in M\) define \[\psi_x(t)=d(x,\gamma(t))-t,\qquad t\ge 0.\] The function \(\psi_x(\cdot)\) is nonincreasing and bounded below by \(-d(x,p)\), and hence the limit \[B^p_v(x):=\lim_{t\to\infty}\bigl(d(x,\gamma(t))-t\bigr)\] exists for every \(x\in M\). This is the Busemann function associated to \(\gamma\), whose basic properties include (see [28] and [29]):

• \(B^p_v:M\to\mathbb{R}\) is \(1\)-Lipschitz;

• \(B^p_v\) is convex, \(C^1\), and satisfies \(\|grad \, B^p_v(x)\|=1\) for all \(x\in M\).

Since they are convex and geometrically explicit on some important Hadamard manifolds, Busemann functions provide natural objective functions or constraint functions for examples and counterexamples in Riemannian optimization. Connections to recent optimization papers can be seen in results by Bento, Cruz Neto, and Melo, who showed that Busemann functions are particularly useful for addressing the absence of nonconstant affine functions on Hadamard manifolds of negative curvature. Their contributions highlight the practical and theoretical suitability of Busemann functions as substitutes for affine functions in formulating and analyzing optimization methods on such manifolds [22], [30], [31]. Because of these features, Busemann functions form a rich, concrete class of convex functions that link global geometric structure with variational and optimization questions on manifolds. Next, we provide an explicit formula for the gradient of Busemann functions on the Poincaré disk, which will be useful to illustrate our method and to suggest directions for future work.

Two unit-speed geodesics \(\gamma\) and \(\alpha\) are called asymptotic if there exists \(C>0\) such that \[d(\gamma(t),\alpha(t))\le C\quad\text{for all }t\ge0.\] Given any unit-speed geodesic \(\beta\) and any \(q\in M\), there exists a unique unit-speed geodesic \(\alpha\) with \(\alpha(0)=q\) that is asymptotic to \(\beta\) (see [32]). A key consequence ([32]) is: for each \(x\in M\) let \(\alpha\) be the unique unit-speed geodesic with \(\alpha(0)=x\) that is asymptotic to the unit-speed geodesic \(\gamma\) where \(\gamma(0) = p\) and \(v=\gamma'(0)\). Then \[grad \, B^p_v(x)=-\alpha'(0).\] In particular, setting \(v_x:=\alpha'(0)\), we have \(\|grad B^p_v(x)\|=1\) and the gradient direction equals the negative of the initial velocity vector of the asymptotic geodesic emanating from \(x\). Let \(\mathbb{D}=\{z\in\mathbb{C}: |z| <1\}\) be the unit disk endowed with the Poincaré metric \[\langle u,v\rangle_{p}=\frac{4\langle u,v\rangle}{(1-|p|^{2})^{2}},\qquad u,v\in T_{p}\mathbb{D},\] where \(\langle\cdot,\cdot\rangle\) denotes the standard inner product. We denote by \(\|u\|\) the norm of the vector \(u\in T_p \mathbb{D}\) in this metric, i.e., \(||u|| = \frac{2|u|}{1-|p|^2}\). With this metric \(\mathbb{D}\) is a Hadamard manifold with constant curvature \(-1\), the Poincaré unit disk \(\mathbb{D}\). For a unit direction \(\eta\in\mathbb{C}\) and the geodesic ray \(\gamma(t)=\eta\tanh(t/2)\) emanating from the origin, the associated Busemann function admits the explicit form \[B^{0}_{\eta}(x)=\ln\!\left(\frac{|x-\eta|^{2}}{1-|x|^{2}}\right),\qquad x\in \mathbb{H}^{2}_{B},\] see, e.g., [28]. This closed form is convenient for explicit computations of horocycles and horoballs. We will now calculate an explicit formula for the gradient of the Busseman function \(B^{0}_{\eta}\). Given \(p \in \mathbb{D}\) and \(\xi \in T_p\mathbb{D}\) with \(||\xi|| = 1\). The Möbius transformation \(\mathcal{M}:\mathbb{D}\to \mathbb{D}\) defined by \[\mathcal{M}(z) = \displaystyle\frac{|\xi|\xi z + |\xi|^2p}{|\xi| \overline{p} \xi z + |\xi|^2},\] is an isometry such that \(\mathcal{M}(0) = p\), see [22]. Observe that \(D\mathcal{M}_{0}(\beta'(0)) = \xi\) where \(\beta(t) = \tanh(t/2)\) in the complex notation. In particular, \((\mathcal{M} \circ \beta)(t)\) is the unique geodesic on the Poincaré disk such that \((\mathcal{M} \circ \beta)(0) = p\) and \((M \circ \beta)'(0) = \xi\), which gives a general expression for the geodesics on the Poincaré disk.

Given \(p \in \mathbb{D}\) let \(c(t)\) the unique geodesic asymptotic to \(\gamma(t)=\eta\tanh(t/2)\) where \(c(0) = p\) and \(c'(0) = \xi\). We can write \[c(t) = \displaystyle\frac{\tanh(t/2)|\xi|\xi + |\xi|^2p}{|\xi| \overline{p} \xi \tanh(t/2) + |\xi|^2}.\] Since \(c\) and \(\gamma\) are asymptotic, it follows that \(\displaystyle\lim_{t \to \infty} c(t) = \eta\) in the Euclidean distance thus

\[\displaystyle\frac{|\xi|\xi + |\xi|^2p}{|\xi| \overline{p} \xi + |\xi|^2} = \eta \implies \xi = \displaystyle\frac{|\xi|(\eta-p)}{1-\eta\overline{p}}.\] On the other hand, \(1 = ||\xi|| = \frac{2|\xi|}{1-|p|^2}\). Hence, \(\xi = \displaystyle\frac{1-|p|^2}{2}\displaystyle\frac{(\eta-p)}{1-\eta\overline{p}}\) and \[grad B^{0}_{\eta}(p) = \displaystyle\frac{1-|p|^2}{2}\displaystyle\frac{(p-\eta)}{1-\eta\overline{p}}.\]

The Figure 1 represents asymptotic geodesics in the Poincaré disk. We will now study a convex function whose properties will be used for further discussion later in the paper. Consider now the geodesic rays \(\alpha(t) = \tanh(-t/2)\), \(\beta(t) = \tanh(t/2)\) and the associated Busemann functions \(B_{\alpha}\) and \(B_{\beta}\) in Poincaré disk. The function \(f:\mathbb{D} \to \mathbb{R}\) defined by \(f(p) = B_{\beta}(p) + B_{\alpha}(p)\) is convex. Note that \(f(0) = 0\), for \(p \neq 0\) let \(\theta\) the angle between \(\exp_0^{-1}(p)\) and \(\beta'(0)\) and \(t = d(0,p)\), where \(d\) denotes the distance in the Poincaré disk. From [22], it follows that \[f(p) = \ln(\cosh t - \sinh t \cos \theta) + \ln(\cosh t + \sinh t \cos \theta).\] Thus, \(f(p) = \ln(1+\sinh^2 t \sin^2 \theta) \geq 0\) and \(f(p) = 0\) only if \(\theta = 0\) or \(\theta = \pi\), and \(S:=\arg\min f\) coincide with the points of the disk on the \(x\)-axis. In particular, \(S\) is unbounded with respect to Poincaré metric. If \(p=qi\) with \(-1 < q <1\) we have \[\begin{align} grad f(p) = \displaystyle\frac{1-q^2}{2}\Big( \displaystyle\frac{1+qi}{1-qi} - \displaystyle\frac{1-qi}{1+qi} \Big)= \displaystyle\frac{2(1-q^2)}{1+q^2}qi. \end{align}\] Therefore, \(-grad \; f(p)\) always points towards the origin for points on the \(y\)-axis.

The following proposition is well known in the Euclidean setting. However, to the best of our knowledge, no proof in the Riemannian setting appears in the literature. For completeness, we provide below a formal statement and proof in the convex case.

Given a point \(x \in M\), a vector \(v \in T_{x}M\) is called a subgradient of \(f\) at \(x\) if the following inequality holds: \[\label{subineq} f(y) \geq f(x) + \langle v, \exp_{x}^{-1}y \rangle, \qquad y \in M.\tag{2}\] The set of all subgradients of \(f\) at \(x\) is denoted by \(\partial f(x)\). Since \(f\) is convex, the set \(\partial f(x)\) is nonempty, compact, and convex for every \(x \in M\). When \(f\) is differentiable at \(x\), the set \(\partial f(x)\) reduces to a single element, namely the gradient vector of \(f\) at \(x\). Moreover, if \(\{z^k\}\) is a bounded sequence and \(v^k \in \partial f(z^k)\) for all \(k \in \mathbb{N}\), then the sequence \(\{v^k\}\) is bounded.

Using the law of cosines for Hadamard manifolds with sectional curvature bounded below (see Proposition 1), we establish the following theorem, which is essential for the proof of the main results of this paper.

The Figure 2 represents the contraction property of the subgradient method on a Riemannian manifold \(M\). It shows the geometric construction of a single Riemannian subgradient step, which we will formalize in the next section: starting from a point \(x \in M\), a subgradient \(g \in \partial f(x)\) is computed in the tangent space \(T_xM\), the normalized descent direction is given by \(s = -g/\|g\|\), and the next iterate is obtained as \(z = \exp_x(\lambda s)\) along the geodesic determined by \(s\). The point \(\bar{x}\) is surrounded by a ball \(B[\bar{x},\delta]\) on which \(f(u) < f(x)\) for all \(u \in B[\bar{x},\delta]\), and the condition \(d(x,\bar{x}) \ge 2\delta\) ensures that \(x\) lies outside this neighborhood. The figure highlights how the update \(z\) moves along the manifold towards the region of lower function values around \(\bar{x}\), illustrating the underlying contraction behavior encoded in the hyperbolic inequality for \(d(z,\bar{x})\).

Throughout the remainder of the paper, unless otherwise stated, \(M\) denotes an \(n\)-dimensional Hadamard manifold with sectional curvature \(\mathrm{sec}_M \geq -\kappa^2\), and \(f\colon M \to \mathbb{R}\) denotes a convex function. With these assumptions in place, we proceed to develop the main results.

3 Subgradient Method↩︎

In this section, we consider the nonsmooth optimization problem \[\min_{x\in M} f(x),\] and, motivated by recent developments and by Shepilov’s classical Euclidean analysis, we study the convergence of the subgradient method on complete Riemannian manifolds with sectional curvature bounded from below. For brevity, detailed assumptions on the step-size sequence are omitted here and stated when needed.

Throughout this section, we use the following notation: \(f^{*} := \inf_{x \in M} f(x)\), \(S := \{ x \in M : f(x) = f^{*} \}\), \(B[\bar{x},\varepsilon] := \{ x \in M : d(x,\bar{x}) \leq \varepsilon \}\), \([f \leq a] := \{ x \in M : f(x) \leq a \}\), and \([f = a] := \{ x \in M : f(x) = a \}\).

3.1 Method↩︎

To define the subgradient method for minimizing \(f\) on \(M\), let \(\{\lambda_k\}\) be a sequence of positive step sizes. The method then generates an iterative sequence \(\{x^k\}\) in \(M\) as follows:

Subgradient Method (SM)
Step 0: Choose \(x^{0} \in M\) and set \(k = 0\).
Step 1: For the current iterate \(x^{k} \in M\), compute a subgradient \(g^{k} \in \partial f(x^{k})\).
Step 2: If \(g^{k} = 0\), STOP. Otherwise, define the next iterate \(x^{k+1} \in M\) by \[\label{method} x^{k+1} := \exp_{x^{k}} \bigl(\lambda_{k} s^{k}\bigr), \qquad s^{k} := -\frac{g^{k}}{\|g^{k}\|}.\tag{12}\]
Step 2: Update \(k \gets k+1\) and return to Step 1.
Since \(f\) is a convex function, the subgradient method is well-defined. Moreover, from inequality (2 ), it follows that \(g^k=0\) if and only if \(x^k \in S\). If there exists an iteration index \({k}_0\) such that \(g^{{k}_0}=0\), then the Subgradient Method is said to be finite. Otherwise, it is said to be infinite.

3.2 Convergence analysis↩︎

In this section we establish convergence results under any of the following conditions: the iterates \(\{x^k\}\) are bounded; the solution set \(S\) is bounded; \(\sum_k \lambda_k^2<\infty\); or \(S\) has nonempty interior. The choice of each case will be specified wherever required.

The following proposition is an immediate consequence of Theorem 3 and it plays a central role in establishing the method’s convergence properties.

In the next two results, we assume that the sequence \(\{\lambda_k\}\) satisfies \(\displaystyle\lim_{k \to \infty} \lambda_k = 0\).

From this point onward, for our convergence analysis, we assume that the sequence \(\{\lambda_k\}\) satisfies: \[\displaystyle\lim_{k \to \infty} \lambda_k = 0 \quad \text{and} \quad \displaystyle\sum_{k=1}^{\infty} \lambda_k = \infty.\] The following result provides a condition under which the method has finite termination.

From now on, we assume that the subgradient method (SM) does not terminate in a finite number of steps, i.e.,\(x^k\notin S\) for all \(k\). The next result shows, in particular, that every accumulation point of \(\{x^k\}\) is a solution of the optimization problem.

Although the next two results have appeared previously in the literature [19], [21], we include them here for completeness and provide alternative proofs that deepen understanding and strengthen the contribution of the present paper centered on Theorem 3, which is instrumental to our analysis and has played an essential role in the development and organization of the paper.

4 Conclusions↩︎

Our main contribution is a Riemannian counterpart of Shepilov’s Euclidean convergence analysis for the subgradient method on manifolds with lower bounded curvature. It complements recent advances in Riemannian convex optimization and, in particular, provides a general tool that illustrates the underlying contraction behavior in the hyperbolic setting which will be useful for the future development of other first-order methods on manifolds. The explicit example constructed in the Poincaré disk model (see Section 2) has an unbounded solution set \(S\). More precisely, \(S:=\arg\min f\) coincides with the points of the disk on the \(x\)-axis. In this example, if we apply the subgradient method starting from an initial point \(x^0=q^0 i\) on the \(y\)-axis then \[x^{k+1}=\exp_{x^{k}}\bigl(\lambda_{k}s^{k}\bigr)=q^{k+1}i\] remains on the \(y\)-axis for all \(k\in\mathbb{N}\). The sequence \(\{x^k\}\) is bounded because \(-\operatorname{grad}f(x^k)\) always points toward the origin, \[d(x^{k+1},0)\le\max\{\lambda_k,d(x^k,0)\},\] and \(\displaystyle\lim_{k\to\infty}\lambda_k=0\). By Theorem 6 we have \(d(x^k,S)\to0\). Since \(d(x^k,S)=d(x^k,0)\), it follows that \(x^k\to0\). Thus, we obtain a situation in which the solution set \(S\) is unbounded, yet the sequence \(\{x^k\}\) remains bounded and converges entirely to an element of \(S\). Although the objective is smooth, the example indicates directions for further theoretical refinement, even in the smooth setting. This construction suggests a broader geometric question. In the comparison arguments used throughout this paper, the constant \(\kappa\) enters only through the geodesic triangles generated by the subgradient step \[x^{k+1}=\exp_{x^k}(\lambda_k s^k), \qquad s^k=-\frac{g^k}{\|g^k\|}.\] Thus, instead of assuming a uniform lower bound \[\sec M\geq -\kappa^2,\] one may ask whether the convergence analysis can be extended to Hadamard manifolds whose curvature is not uniformly bounded below, but whose negative part grows at most linearly with the distance from a fixed reference point \(\bar x\in M\).

More precisely, a natural condition is the existence of constants \(A,B\geq 0\) such that, for every \(p\in M\) and every two-dimensional subspace \(\sigma\subset T_pM\), \[\label{eq:LC} K_p(\sigma)\geq -\bigl(A+B\,d(p,\bar x)\bigr).\tag{28}\]

The case \(B=0\) recovers the constant lower curvature bound considered in the present work, with \(\kappa=\sqrt{A}\). If a geodesic triangle associated with a subgradient step is contained in a ball \(B[\bar x,R]\), then condition 28 implies the local estimate \[K_p(\sigma)\geq -(A+BR),\] in that region. Consequently, the same hyperbolic comparison estimates may be applied locally with the effective parameter \[\kappa_R:=\sqrt{A+BR}.\] This observation suggests a possible extension of the present analysis. The main difficulty is that the effective comparison parameter, \(\kappa_R\), depends on the size of the region visited by the iterates. Therefore, one would need a bootstrap argument showing that the curvature contribution generated along the orbit remains controlled before boundedness of the sequence is known. Such an approach would extend the current theory from manifolds with uniformly bounded below curvature to a class of Hadamard manifolds with linearly decaying lower curvature bounds, and will be investigated in future work.

References↩︎