Isaac Kaminer

Abram H. Clark, Liraz Mudrik, Colton Kawamura et al.

Designing autonomous drone swarms is hampered by a vast design space spanning platform, algorithmic, and numerical-strength choices. We perform large-scale agent-based simulations in three canonical scenarios: swarm-on-swarm battle, cooperative area search with attrition, and pursuit of scattering targets. We demonstrate how dimensional-analysis and data-scaling can be leveraged to collapse performance data onto scaling functions that are mathematically simple, yet counterintuitive and therefore difficult to predict a priori. These scaling laws reveal success-failure boundaries, including sharp break points which we show can be framed as an ``effective swarm size.'' Additionally, we show how this technique can be used to quantify trade-offs between agent count and platform parameters such as velocity, sensing or weapon range, and attrition rate. Furthermore, we show the benefits of embedding an optimal path planning loop within this framework, which can qualitatively improve the scaling laws that govern the outcome. The methods we demonstrate are highly flexible and would enable rapid, budget-aware sizing and algorithm selection for large autonomous swarms.

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.

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.

Donald W. Peltier, Isaac Kaminer, Abram Clark et al.

Understanding the characteristics of swarming autonomous agents is critical for defense and security applications. This article presents a study on using supervised neural network time series classification (NN TSC) to predict key attributes and tactics of swarming autonomous agents for military contexts. Specifically, NN TSC is applied to infer two binary attributes - communication and proportional navigation - which combine to define four mutually exclusive swarm tactics. We identify a gap in literature on using NNs for swarm classification and demonstrate the effectiveness of NN TSC in rapidly deducing intelligence about attacking swarms to inform counter-maneuvers. Through simulated swarm-vs-swarm engagements, we evaluate NN TSC performance in terms of observation window requirements, noise robustness, and scalability to swarm size. Key findings show NNs can predict swarm behaviors with 97% accuracy using short observation windows of 20 time steps, while also demonstrating graceful degradation down to 80% accuracy under 50% noise, as well as excellent scalability to swarm sizes from 10 to 100 agents. These capabilities are promising for real-time decision-making support in defense scenarios by rapidly inferring insights about swarm behavior.

Donald W. Peltier, Isaac Kaminer, Abram Clark et al.

Having the ability to infer characteristics of autonomous agents would profoundly revolutionize defense, security, and civil applications. Our previous work was the first to demonstrate that supervised neural network time series classification (NN TSC) could rapidly predict the tactics of swarming autonomous agents in military contexts, providing intelligence to inform counter-maneuvers. However, most autonomous interactions, especially military engagements, are fraught with uncertainty, raising questions about the practicality of using a pretrained classifier. This article addresses that challenge by leveraging expected operational variations to construct a richer dataset, resulting in a more robust NN with improved inference performance in scenarios characterized by significant uncertainties. Specifically, diverse datasets are created by simulating variations in defender numbers, defender motions, and measurement noise levels. Key findings indicate that robust NNs trained on an enriched dataset exhibit enhanced classification accuracy and offer operational flexibility, such as reducing resources required and offering adherence to trajectory constraints. Furthermore, we present a new framework for optimally deploying a trained NN by the defenders. The framework involves optimizing defender trajectories that elicit adversary responses that maximize the probability of correct NN tactic classification while also satisfying operational constraints imposed on the defenders.

Venanzio Cichella, Isaac Kaminer, Claire Walton et al.

Bernstein polynomial approximation to a continuous function has a slower rate of convergence as compared to other approximation methods. "The fact seems to have precluded any numerical application of Bernstein polynomials from having been made. Perhaps they will find application when the properties of the approximant in the large are of more importance than the closeness of the approximation." -- has remarked P.J. Davis in his 1963 book Interpolation and Approximation. This paper presents a direct approximation method for nonlinear optimal control problems with mixed input and state constraints based on Bernstein polynomial approximation. We provide a rigorous analysis showing that the proposed method yields consistent approximations of time continuous optimal control problems. Furthermore, we demonstrate that the proposed method can also be used for costate estimation of the optimal control problems. This latter result leads to the formulation of the Covector Mapping Theorem for Bernstein polynomial approximation. Finally, we explore the numerical and geometric properties of Bernstein polynomials, and illustrate the advantages of the proposed approximation method through several numerical examples.