Zeroth-order optimization on Riemannian manifolds is relevant when objectives are available only through function evaluations, but conventional tangent estimators can be unstable under non-Gaussian oracle errors. We propose PRISM-ZO, a projection-robust framework that samples low-dimensional random tangent subspaces and combines symmetric finite differences with median-of-means or Huber aggregation. We establish the unbiasedness of the correctly rescaled projected direction in expectation over the random subspace, quantify its conditional projection error, prove an explicit expected bound for convergence to an approximate first-order stationary point under a stated estimator-error model, and give a verifiable condition for detecting projected negative curvature. In sparse PCA experiments with 10 random seeds, PRISM-ZO variants are compared with full-dimensional zeroth-order and first-order power baselines under Gaussian, Student-t and Cauchy perturbations; the results report means, standard deviations and 95% confidence intervals and identify both beneficial and failure regimes. The hard-thresholded sparse update is treated as an empirical proxy rather than a smooth retraction.

Urban taxi systems are continuously exposed to recurrent peak-hour demand surges that impose sustained operational stress, while existing resilience studies have predominantly focused on rare extreme disruptions. This study develops a coupled spatiotemporal–network resilience framework to investigate how urban taxi systems respond to routine, high-frequency peak-hour disturbances. Based on two weeks of high-resolution taxi trip data collected in Zhuhai, eight representative daily disturbance events are identified through a data-driven detection approach capturing distinct commuting and leisure travel demand regimes. Spatiotemporal resilience and network resilience are assessed separately considering resistance and adaptability at both node and system levels. These dimensions are subsequently integrated through an entropy-weighted coupling method to reveal multidimensional resilience states. Results show pronounced scenario-dependent resilience patterns. Structured commuting peaks maintain moderate operational performance despite limited structural flexibility, whereas leisure-oriented disturbances exhibit higher network robustness and greater route diversity, facilitating adaptation despite more dispersed operational demands. For all identified events, resilience is determined not by total demand volume, but by spatiotemporal distribution of trips and the alignment between operational performance and network topology. The coupled framework further uncovers scale-dependent effects. Spatiotemporal resilience exhibits significant spatial variations across disturbance scenarios, whereas network resilience remains comparatively stable owing to persistent structural properties. High-demand nodes often exhibit stronger integrated resilience owing to structural centrality, while system-level resilience is governed by demand dispersion and flow organization. By explicitly linking spatiotemporal operations to network topology under recurrent daily disturbances, this study lays a diagnostic foundation for scenario-adaptive resilience management in routine urban transportation operations.

Many robotic applications—from drone navigation to surgical automation—require planning trajectories that respect directional constraints, which are inherently non-Euclidean. Trajectory planning in robotics often involves orientation or directional constraints that naturally reside on Riemannian manifolds. In particular, when robot motion is governed by orthogonality constraints—such as in directional alignment, pose smoothing, or coordinated orientation—the optimization variables lie on the Stiefel manifold. Traditional Euclidean optimization methods struggle with such constraints, leading to suboptimal or infeasible trajectories. This article proposes a Riemannian optimization framework tailored to the Stiefel manifold for robot trajectory planning. By leveraging the intrinsic geometry of the manifold, we design efficient algorithms that compute gradient updates via Riemannian geometry and enforce feasibility through retraction mappings. The proposed method enables smooth and constraint-respecting trajectory generation, particularly in scenarios where robot poses or motion directions are orthonormal by design. We validate our approach through simulated path planning tasks involving orientation-aware constraints. Compared to baseline projection-based or unconstrained methods, our approach achieves better convergence behavior, improved geometric consistency, and enhanced applicability in robotics. The results demonstrate that manifold-aware optimization not only improves theoretical soundness but also provides practical benefits for engineering systems.

Recently, the adaptive regularized proximal quasi-Newton (ARPQN) method has demonstrated a strong performance in solving composite optimization problems over the Stiefel manifold. However, its reliance on first-order information limits its applicability to scenarios where gradient and Hessian evaluations are unavailable or costly. In this paper, we propose a zeroth-order adaptive regularized proximal quasi-Newton method (ZO-ARPQN) for black-box composite optimization over Riemannian manifolds, particularly the Stiefel and symmetric positive definite (SPD) manifolds. The proposed method estimates the Riemannian gradient and curvature information through randomized one-point finite-difference approximations and adaptively updates a regularized quasi-Newton matrix to capture the local manifold geometry. Theoretically, we established global convergence and complex analyses under mild assumptions. More importantly, by incorporating curvature-aware regularization and random perturbations in the proximal quasi-Newton framework, we proved that ZO-ARPQN can escape strict saddle points with a high probability. This guarantees convergence to a stationary point, even in the absence of explicit gradients. Extensive numerical experiments were conducted on manifold-constrained problems, including sparse PCA and robot stiffness tuning. These demonstrated that ZO-ARPQN shows a competitive convergence behavior compared with other state-of-the-art Riemannian optimization methods, while requiring only function evaluations.

This paper proposes a Distributed Adaptive Regularization Algorithm (DARN) for solving composite non-convex and non-smooth optimization problems in multi-agent systems. The algorithm employs a three-phase iterative framework to achieve efficient collaborative optimization: (1) a local regularized optimization step, which utilizes proximal mappings to enforce strong convexity of weakly convex objectives and ensure subproblem well-posedness; (2) a consensus update based on doubly stochastic matrices, guaranteeing asymptotic convergence of agent states to a global consensus point; and (3) an innovative adaptive regularization mechanism that dynamically adjusts regularization strength using local function value variations to balance stability and convergence speed. Theoretical analysis demonstrates that the algorithm maintains strict monotonic descent under non-convex and non-smooth conditions by constructing a mixed time-scale Lyapunov function, achieving a sublinear convergence rate. Notably, we prove that the projection-based update rule for regularization parameters preserves lower-bound constraints, while spectral decay properties of consensus errors and perturbations from local updates are globally governed by the Lyapunov function. Numerical experiments validate the algorithm’s superiority in sparse principal component analysis and robust matrix completion tasks, showing a 6.6% improvement in convergence speed and a 51.7% reduction in consensus error compared to fixed-regularization methods. This work provides theoretical guarantees and an efficient framework for distributed non-convex optimization in heterogeneous networks.

This paper proposes a stochastic differential equation (SDE)-based global optimization method on matrix Lie groups, specifically addressing the challenge of local optima trapping in optimization over special orthogonal groups SO(n) and related structures. By combining Riemannian conjugate gradient methods with an intermittent diffusion mechanism, we develop an optimization framework that preserves geometric structures while maintaining global exploration capabilities. Theoretical analysis demonstrates first-order convergence between the continuous-time SDE and its discrete numerical implementation, with almost sure convergence to the global optimum under appropriate conditions. Experimental validation through two applications—multimodal function optimization and robotic path planning—confirms the effectiveness of the proposed ID-RCG algorithm, showing superior performance in both convergence rate and global search capability compared to conventional methods.