Sean Kragelund

Liraz Mudrik, Sean Kragelund, Isaac Kaminer

Efficient trajectory generation is crucial for autonomous systems; however, current numerical methods often struggle to handle periodic behaviors effectively, particularly when the onboard sensors require equidistant temporal sampling. This paper introduces a novel mixed Bernstein-Fourier approximation framework tailored explicitly for optimal motion planning. Our proposed methodology leverages the uniform convergence properties of Bernstein polynomials for nonperiodic behaviors while effectively capturing periodic dynamics through the Fourier series. Theoretical results are established, including uniform convergence proofs for approximations of functions, derivatives, and integrals, as well as detailed error bound analyses. We further introduce a regulated least squares approach for determining approximation coefficients, enhancing numerical stability and practical applicability. Within an optimal control context, we establish the feasibility and consistency of approximated solutions to their continuous counterparts. We also extend the covector mapping theorem, providing theoretical guarantees for approximating dual variables crucial in verifying the necessary optimality conditions from Pontryagin's Maximum Principle. Numerical examples illustrate the method's superior performance, demonstrating substantial improvements in computational efficiency and precision in scenarios with complex periodic constraints and dynamics. Our mixed Bernstein-Fourier methodology thus presents a robust, theoretically grounded, and computationally efficient approach for advanced optimal trajectory planning in autonomous systems.

Liraz Mudrik, Isaac Kaminer, Sean Kragelund et al.

This paper proposes the first fully distributed algorithm for finding the Generalized Nash Equilibrium (GNE) of a non-cooperative game with shared coupling constraints and general cost coupling at a user-prescribed finite time T. As a foundation, a centralized gradient-based prescribed-time convergence result is established for the GNE problem, extending the optimization Lyapunov function framework to gradient dynamics, the only known realization among existing alternatives that naturally decomposes into per-agent computations. Building on this, a fully distributed architecture is designed in which each agent concurrently runs three coupled dynamics: a prescribed-time distributed state observer, a gradient-based optimization law, and a dual consensus mechanism that enforces the shared-multiplier requirement of the variational GNE, thus guaranteeing convergence to the same solution as the centralized case. The simultaneous operation of these layers creates bidirectional perturbations between consensus and optimization, which are resolved through gain synchronization that matches the temporal singularities of the optimization and consensus layers, ensuring all error components vanish exactly at T. The Fischer-Burmeister reformulation renders the algorithm projection-free and guarantees constraint satisfaction at the deadline. Numerical simulations on a Nash-Cournot game and a time-critical sensor coverage problem validate the approach.

Liraz Mudrik, Isaac Kaminer, Sean Kragelund et al.

Nonconvex optimization underlies many modern machine learning and control tasks, where saddle points pose the dominant obstacle to reliable convergence in high-dimensional settings. Escaping these saddle points deterministically and at a controllable rate remains an open challenge: gradient descent is blind to curvature, stochastic perturbation methods lack deterministic guarantees, and Newton-type approaches suffer from Hessian singularity. We present Curvature-Regularized Gradient Dynamics (CRGD), which augments the objective with a smooth penalty on the most negative Hessian eigenvalue, yielding an augmented cost that serves as an optimization Lyapunov function with user-selectable convergence rates to second-order stationary points. Numerical experiments on a nonconvex matrix factorization example confirm that CRGD escapes saddle points across all tested configurations, with escape time that decreases with the eigenvalue gap, in contrast to gradient descent, whose escape time grows inversely with the gap.

Liraz Mudrik, Isaac Kaminer, Sean Kragelund et al.

Optimization plays a central role in intelligent systems and cyber-physical technologies, where speed and reliability of convergence directly impact performance. In control theory, optimization-centric methods are standard: controllers are designed by repeatedly solving optimization problems, as in linear quadratic regulation, $H_\infty$ control, and model predictive control. In contrast, this paper develops a control-centric framework for optimization itself, where algorithms are constructed directly from Lyapunov stability principles rather than being proposed first and analyzed afterward. A key element is the stationarity vector, which encodes first-order optimality conditions and enables Lyapunov-based convergence analysis. By pairing a Lyapunov function with a selectable decay law, we obtain continuous-time dynamics with guaranteed exponential, finite-time, fixed-time, or prescribed-time convergence. Within this framework, we introduce three feedback realizations of increasing restrictiveness: the Hessian-gradient, Newton, and gradient dynamics. Each realization shapes the decay of the stationarity vector to achieve the desired rate. These constructions unify unconstrained optimization, extend naturally to constrained problems via Lyapunov-consistent primal-dual dynamics, and broaden the results for minimax and generalized Nash equilibrium seeking problems beyond exponential stability. The framework provides systematic design tools for optimization algorithms in control and game-theoretic problems.