variational methods

This paper investigates necessary conditions for optimal Markovian control policies in finite-horizon stochastic control problems. Departing from conventional approaches based on dynamic programming or the stochastic maximum principle, it systematically extends classical calculus of variations to stochastic settings—integrating Itô calculus and stochastic analysis to rigorously derive first-order variational conditions for optimality, namely a stochastic Euler–Lagrange-type equation. Crucially, this framework does not require differentiability of the value function, thereby providing a novel necessity analysis tool for nonlinear and nonconvex stochastic control problems. As a key validation, the paper solves the classical Merton portfolio optimization problem analytically; the resulting explicit optimal policy coincides exactly with established results, confirming both the mathematical rigor and practical efficacy of the proposed method.

This work proposes a novel paradigm termed “Optimized Natural Physics” to investigate whether optimization algorithms adhere to natural laws of motion induced by the objective function. By establishing equivalence between optimal control problems and generalized KKT conditions, the authors construct a natural vector field governed by non-Newtonian dynamics. Leveraging Pontryagin’s minimum principle, Hamilton–Jacobi inequalities, and energy dissipation mechanisms, they design control strategies possessing inverse optimality. This framework not only unifies the interpretation of diverse existing optimization algorithms but also enables the systematic derivation of new ones. The approach demonstrates that global optimization can be achieved through deliberate modulation of jumps and dissipation, thereby providing a physically intuitive and mathematically unified foundation for optimization theory.

This work addresses discrete-time nonlinear optimal control problems by unifying classical algorithms—including gradient descent, Gauss–Newton, Newton’s method, and differential dynamic programming (DDP)—within a differentiable programming framework. Methodologically, it introduces the first modular, end-to-end differentiable algorithm template library built upon linear/quadratic approximations (e.g., LQR), enabled by automatic differentiation. Theoretically, it provides a unified derivation of computational complexity and sufficient optimality conditions across all methods. Practically, it incorporates adaptive line search and regularization strategies, and validates efficacy on benchmark tasks such as autonomous racing with a bicycle model. All implementations are open-sourced, demonstrating both efficient gradient propagation and strong generalization across diverse control problems.

This work addresses the common omission of fundamental physical constraints—such as inertia, gravity, and viscous drag—in robot motion optimization. We propose a unified optimal control framework grounded in differential geometry. Methodologically, we model viscous drag as a Riemannian metric on the configuration manifold, thereby unifying kinetic energy and gravitational potential fields, and derive geometric optimal control equations incorporating curvature effects. Indirect optimal control is solved via Lagrangian mechanics and manifold-based variational calculus. Experiments on a two-link planar manipulator and a UR5 robot demonstrate that the proposed model substantially alters optimal trajectory shape and energy distribution, enhancing both physical realizability and energy efficiency. Our core contribution is a novel geometric modeling paradigm that synergistically integrates drag, curvature, and potential fields—establishing a new theoretical foundation for physics-informed robotic trajectory optimization.

Modeling strongly path-dependent financial derivatives—such as exotic options and variance instruments—remains challenging due to the non-Markovian nature of path-dependent functionals. Method: This paper introduces the “occupied process” framework, augmenting the original process $X$ with its occupation measure flow $O$ to form a Markovian lifted system $(O,X)$. It defines the novel “occupation derivative”, unifying functional Itô calculus and mean-field derivatives, and recasts a broad class of path-dependent PDEs as parabolic equations in the occupation measure time variable. Contribution/Results: The framework enables an Itô calculus tailored to path occupation-time functionals and extends the Feynman–Kac formula accordingly. It yields closed-form solutions to local-time-driven optimal stopping problems, with direct applications to corridor variance swap pricing and path-dependent volatility modeling. By bridging stochastic analysis, mean-field theory, and financial mathematics, this work substantially expands both the theoretical foundations and practical applicability of path-dependent stochastic modeling.

This work addresses the lack of a unified framework in existing optimizer design, which often relies on heuristic modifications and struggles to balance stability and generalization. The authors propose the first systematic approach that integrates control theory with Riemannian geometry, modeling the optimization process as a discrete-time controlled dynamical system on a Riemannian manifold. By introducing normally attracting invariant manifolds (NAIMs) and strict Lyapunov functions, they establish a theoretically grounded framework for generating optimizers with provable convergence guarantees. This framework not only recovers classical algorithms but also yields novel optimizers that achieve state-of-the-art performance on large-scale benchmarks. Geometric diagnostics further validate the method’s efficacy, offering a stable, interpretable, and theoretically rigorous toolkit for optimizer design.

This work proposes a unified framework based on stochastic control and mean-field theory for solving global optimization problems in both Euclidean space and the Wasserstein space of probability measures. By introducing a regularized stochastic control problem and leveraging dynamic programming, the Cole–Hopf transformation, and the Feynman–Kac formula, the authors derive a computable approximation to the original problem. A system of N interacting particles is then employed to numerically solve the associated measure-valued dynamics. Theoretical analysis establishes that as the regularization parameter vanishes and the number of particles tends to infinity, the value function of the control problem converges to the global minimum of the original optimization problem. Numerical experiments confirm the method’s efficacy and validate the predicted theoretical convergence rates. This study represents the first integration of stochastic control with mean-field theory, offering a novel paradigm and rigorous theoretical guarantees for global optimization across both spaces.

This study addresses the non-smooth, constrained user utility maximization problem inherent in Perturbed Utility Route Choice (PURC) models by introducing a unified convex duality framework. The proposed approach transforms the original problem into an unconstrained, differentiable concave maximization task, enabling efficient gradient-based optimization. By leveraging the convex conjugate of link-specific perturbation functions, the method uniquely recovers optimal route flows link-by-link. This work establishes, for the first time, a rigorous convex duality theory for PURC models, revealing a structural analogy to electrical current flows. The framework facilitates rapid sensitivity analysis and scalable computation, significantly enhancing both efficiency and applicability for real-time solution and parameter sensitivity evaluation in large-scale, complex transportation networks.

This work addresses the problem of feedback motion planning for continuous-time stochastic nonlinear systems under Signal Temporal Logic (STL) specifications by proposing a novel framework that integrates predicate erosion with probabilistic reachable tubes. Predicate erosion is employed to transform stochastic STL constraints into tightened deterministic ones, while probabilistic reachable tubes quantify the deviation of stochastic trajectories from their nominal counterparts. Leveraging contraction theory, a tracking controller is designed to establish a closed-loop planning pipeline. The proposed approach significantly reduces the conservatism inherent in conventional methods, achieving high STL satisfaction probability without compromising planning performance. Simulations and real-world experiments on a quadrupedal robot demonstrate that the method outperforms baseline approaches in both STL satisfaction rate and computational efficiency.