Optimization and Control (math.OC)

Abstract: In nature, limbless locomotion is adopted by a wide range of organisms at various length scales. Interestingly, undulatory, crawling and inching/looping gait constitutes a fundamental class of limbless locomotion and is often observed in many species such as caterpillars, earthworms, leeches, larvae, and \emph{C. elegans}, to name a few. In this work, we developed a computationally efficient 3D Finite Element (FE) based unified framework for the locomotion of limbless organisms on soft substrates. Muscle activity is simulated with a multiplicative decomposition of deformation gradient, which allows mimicking a broad range of locomotion patterns in 3D solids on frictional substrates. In particular, a two-field FE formulation based on positions and velocities is proposed. Governing partial differential equations are transformed into equivalent time-continuous differential-algebraic equations (DAEs). Next, the optimal locomotion strategies are studied in the framework of optimal control theory. We resort to adjoint-based methods and deduce the first-order optimality conditions, that yield a system of DAEs with two-point end conditions. Hidden symplectic structure and Symplectic Euler time integration of optimality conditions have been discussed. The resulting discrete first-order optimality conditions form a non-linear programming problem that is solved efficiently with the Forward Backwards Sweep Method. Finally, some numerical examples are provided to demonstrate the comprehensiveness of the proposed computational framework and investigate the energy-efficient optimal limbless locomotion strategy out of distinct locomotion patterns adopted by limbless organisms.

Abstract: Linear fusion is a cornerstone of estimation theory. Optimal linear fusion was derived by Bar-Shalom and Campo in the 1980s. It requires knowledge of the cross-covariances between the errors of the estimators. In distributed or cooperative systems, these cross-covariances are difficult to compute. To avoid an underestimation of the errors when these cross-covariances are unknown, conservative fusions must be performed. A conservative fusion provides a fused estimator with a covariance bound which is guaranteed to be larger than the true (but not computable) covariance of the error. Previous research by Reinhardt et al. proved that, if no additional assumption is made about the errors of the estimators, the minimal bound for fusing two estimators is given by a fusion called Covariance Intersection (CI). In practice, the errors of the estimators often have an uncorrelated component, because the dynamic or measurement noise is assumed to be independent. In this context, CI is no longer the optimal method and an adaptation called Split Covariance Intersection (SCI) has been designed to take advantage from these uncorrelated components. The contribution of this paper is to prove that SCI is the optimal fusion rule for two estimators under the assumption that they have an uncorrelated component. It is proved that SCI provides the optimal covariance bound with respect to any increasing cost function. To prove the result, a minimal volume that should contain all conservative bounds is derived, and the SCI bounds are proved to be the only bounds that tightly circumscribe this minimal volume.

Abstract: This paper studies the distributed optimization problem over directed networks with noisy information-sharing. To resolve the imperfect communication issue over directed networks, a series of noise-robust variants of Push-Pull/AB method have been developed. These methods improve the robustness of Push-Pull method against the information-sharing noise through adding small factors on weight matrices and replacing the global gradient tracking with the cumulative gradient tracking. Based on the two techniques, we propose a new variant of the Push-Pull method by presenting a novel mechanism of inter-agent information aggregation, named variance-reduced aggregation (VRA). VRA helps us to release some conditions on the objective function and networks. When the objective function is convex and the sharing-information noise is variance-unbounded, it can be shown that the proposed method converges to the optimal solution almost surely. When the objective function is strongly convex and the sharing-information noise is variance-bounded, the proposed method achieves the convergence rate of $\mathcal{O}\left(k^{-(1-\epsilon)}\right)$ in the mean square sense, where $\epsilon$ could be close to 0 infinitely. Simulated experiments on ridge regression problems verify the effectiveness of the proposed method.

Abstract: In this short note we provide a proof of boundedness of solutions for a network system composed of heterogeneous nonlinear autonomous systems interconnected over a directed graph. The sole assumptions imposed are that the systems are semi-passive [1] and the graph contains a spanning tree.

Abstract: We consider dynamic games defined over an infinite horizon, characterized by linear, discrete-time dynamics and quadratic cost functionals. Considering such linear-quadratic (LQ) dynamic games, we focus on their solutions in terms Nash equilibrium strategies. Both Feedback (F-NE) and Open-Loop (OL-NE) Nash equilibrium solutions are considered. The contributions of the paper are threefold. First, our detailed study reveals some interesting structural insights in relation to F-NE solutions. Second, as a stepping stone towards our consideration of OL-NE strategies, we consider a specific infinite-horizon discrete-time (single-player) optimal control problem, wherein the dynamics are influenced by a known exogenous input and draw connections between its solution obtained via Dynamic Programming and Pontryagin's Minimum Principle. Finally, we exploit the latter result to provide a characterization of OL-NE strategies of the class of infinite-horizon dynamic games. The results and key observations made throughout the paper are illustrated via a numerical example.

Abstract: Decentralized optimization is typically studied under the assumption of noise-free transmission. However, real-world scenarios often involve the presence of noise due to factors such as additive white Gaussian noise channels or probabilistic quantization of transmitted data. These sources of noise have the potential to degrade the performance of decentralized optimization algorithms if not effectively addressed. In this paper, we focus on the noisy communication setting and propose an algorithm that bridges the performance gap caused by communication noise while also mitigating other challenges like data heterogeneity. We establish theoretical results of the proposed algorithm that quantify the effect of communication noise and gradient noise on the performance of the algorithm. Notably, our algorithm achieves the optimal convergence rate for minimizing strongly convex, smooth functions in the context of inexact communication and stochastic gradients. Finally, we illustrate the superior performance of the proposed algorithm compared to its state-of-the-art counterparts on machine learning problems using MNIST and CIFAR-10 datasets.