Control Theory and Dynamical Systems

2024

1. 

   Accelerated Saddle Flow Dynamics for Bilinearly Coupled Minimax Problems

   Yingzhu Liu, Enrique Mallada, Zhongkui Li, and Pengcheng You

   2024

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   Minimax problems have attracted much attention due to various applications in constrained optimization problems and zero-sum games. Identifying saddle points within these problems is crucial, and saddle flow dynamics offer a straightforward yet useful approach. This study focuses on a class of bilinearly coupled minimax problems with strongly convex-linear objective functions. We design an accelerated algorithm based on saddle flow dynamics, achieving a convergence rate beyond the stereotype limit (the strong convexity constant). The algorithm is derived from a sequential two-step transformation of a given objective function. First, a change of variables is applied to render the objective function better conditioned, introducing strong concavity (from linearity) while preserving strong convexity. Second, proximal regularization, when staggered with the first step, further enhances the strong convexity of the objective function by shifting some of the obtained strong concavity. After these transformations, saddle flow dynamics based on the new objective function can be tuned for accelerated exponential convergence. Besides, such an approach can be extended to weakly convex-weakly concave functions and still guarantees exponential convergence to one stationary point. The theory is verified by a numerical test on an affine equality-constrained convex optimization problem.

2. 

   A Unified Analysis of Saddle Flow Dynamics: Stability and Algorithm Design

   Pengcheng You, Yingzhu Liu, and Enrique Mallada

   2024

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   This work examines the conditions for asymptotic and exponential convergence of saddle flow dynamics of convex-concave functions. First, we propose an observability-based certificate for asymptotic convergence, directly bridging the gap between the invariant set in a LaSalle argument and the equilibrium set of saddle flows. This certificate generalizes conventional conditions for convergence, e.g., strict convexity-concavity, and leads to a novel state-augmentation method that requires minimal assumptions for asymptotic convergence. We also show that global exponential stability follows from strong convexity-strong concavity, providing a lower-bound estimate of the convergence rate. This insight also explains the convergence of proximal saddle flows for strongly convex-concave objective functions. Our results generalize to dynamics with projections on the vector field and have applications in solving constrained convex optimization via primal-dual methods. Based on these insights, we study four algorithms built upon different Lagrangian function transformations. We validate our work by applying these methods to solve a network flow optimization and a Lasso regression problem.

Convex and Nonconvex Optimization

Data-driven Optimization and Control

2025

1. 

   Discovering Mechanistic Causality from Time Series: A Behavioral-System Approach

   Yingzhu Liu, Shengyuan Huang, Zhongkui Li, Xiaoguang Yang, and Wenjun Mei

   May 2025

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   Identifying “true causality” is a fundamental challenge in complex systems research. Widely adopted methods, like the Granger causality test, capture statistical dependencies between variables rather than genuine driver-response mechanisms. This critical gap stems from the absence of mathematical tools that reliably reconstruct underlying system dynamics from observational time-series data. In this paper, we introduce a new control-based method for causality discovery through the behavior-system theory, which represents dynamical systems via trajectory spaces and has been widely used in data-driven control. Our core contribution is the \textbfBehavior-\textbfenabled \textbfCausality test (the BeCaus test), which transforms causality discovery into solving fictitious control problems. By exploiting the intrinsic asymmetry between system inputs and outputs, the proposed method operationalizes our conceptualization of mechanistic causality: variable X is a cause of Y if X (partially) drives the evolution of Y. We establish conditions for linear time-invariant systems to be causality-discoverable, i.e., conditions for the BeCaus test to distinguish four basic causal structures (independence, full causality, partial causality, and latent-common-cause relation). Notably, our approach accommodates open systems with unobserved inputs. Moreover, an exploratory case study indicates the new method’s potential extensibility to nonlinear systems.