arXiv subjects
Publications and source records attributed to Huanshui Zhang.
This paper proposes a novel second-order optimization algorithm based on the Optimal Control Principle (OCP), applicable to large-scale optimization problems in neural network training. The algorithm has a computational complexity of O(d) and strong robustness. Extensive experiments on multiple benchmarks demonstrate the significant superiority of the proposed method.
In this paper, the optimal consensus problem for general nonlinear multi-agent systems is studied, where both leaderless and leader-follower cases are considered in a unified framework. The key idea is to convert consensus problems into optimal control problems where the objective of each agent with nonlinear dynamics is to design the control input minimizing the global consensus cost function. Compared with the existing distributed consensus control for nonlinear multi-agent systems, we propose a distributed optimal consensus algorithm based on the optimal control principle (OCP) method, and two enhanced algorithms are developed under the model predictive control (MPC) framework,these two algorithms demonstrate broader applicability when handling general nonlinear multi-agent systems. Moreover, the convergence and superlinear convergence rate of the proposed algorithms are rigorously analyzed. Numerical simulations demonstrate the effectiveness of the proposed algorithms.
This paper studies open-loop and feedback solutions to leader-follower mean field linear-quadratic-Gaussian games with multiplicative noise by the direct approach. The leader-follower game involves a leader and many followers, where the state and control weight matrices in their costs are not limited to be positive definite. From variational analysis with mean field approximations, we obtain a set of open-loop controls in terms of solutions to mean field forward-backward stochastic differential equations. By applying the matrix maximum principle, a set of decentralized feedback strategies is constructed. Distinct from traditional works, a cross term has appeared in derivation due to the presence of mean field terms. For open-loop and feedback solutions, the corresponding optimal costs of all players are explicitly given in terms of the solutions to two Riccati equations, respectively.
In this paper, the problem of distributively seeking the equilibria of aggregative games with bilevel structures is studied. Different from the traditional aggregative games, here the aggregation is determined by the minimizer of a virtual leader's objective function in the inner level, which depends on the actions of the players in the outer level. Moreover, the global objective function of the virtual leader is formed by the sum of some local functions with two arguments, each of which is strongly convex with respect to the second argument. When making decisions, each player in the outer level only has access to a local part of the virtual leader's objective function. To handle this problem, first, we propose a second order gradient-based distributed algorithm, where the Hessian matrices associated with the objective functions of the leader are involved. By the algorithm, players update their actions while cooperatively minimizing the objective function of the virtual leader to estimate the aggregation by communicating with their neighbors via a connected graph. Under mild assumptions on the graph and cost functions, we prove that the actions of players asymptotically converge to the Nash equilibrium point. Then, for the case where the Hessian matrices associated with the objective functions of the virtual leader are not available, we propose a first order gradient-based distributed algorithm, where a distributed two-point estimate strategy is developed to estimate the gradients of players' cost functions in the outer level. Under the same conditions, we prove that the convergence errors of players' actions to the Nash equilibrium point are linear with respect to the estimate parameters. Finally, simulations are provided to demonstrate the effectiveness of our theoretical results.
This paper addresses the visual localization problem in Global Positioning System (GPS)-denied environments, where computational resources are often limited. To achieve efficient and robust performance under these constraints, we propose a novel algorithm. The algorithm stems from the optimal control principle (OCP). It incorporates diagonal information estimation of the Hessian matrix, which results in training a higher-performance deep neural network and accelerates optimization convergence. Experimental results on public datasets demonstrate that the final model achieves competitive localization accuracy and exhibits remarkable generalization capability. This study provides new insights for developing high-performance offline positioning systems.
This paper considers distributed optimization problems, where each agent cooperatively minimizes the sum of local objective functions through the communication with its neighbors. The widely adopted distributed gradient method in solving this problem suffers from slow convergence rates, which motivates us to incorporate the second-order information of the objective functions. However, the challenge arises from the unique structure of the inverse of the Hessian matrix, which prevents the direct distributed implementation of the second-order method. We overcome this challenge by proposing a novel optimization framework. The key idea is to transform the distributed optimization problem into an optimal control problem. Using Pontryagin's maximum principle and the associated forward-backward difference equations (FBDEs), we derive a new distributed optimization algorithm that incorporates the second-order information without requiring the computation of the inverse of the Hessian matrix. Furthermore, the superlinear convergence of the proposed algorithm is proved under some mild assumptions. Finally, we also propose a variant of the algorithm to balance the number of iterations and communication.
This paper delves into studying the differences and connections between open-loop and closed-loop strategies for the linear quadratic (LQ) mean field games (MFGs) by the direct approach. The investigation begins with the finite-population system for solving the solvability of open-loop and closed-loop systems within a unified framework under the global information pattern. By a comprehensive analysis through variational methods, the necessary and sufficient conditions are obtained for the existence of centralized open-loop and closed-loop Nash equilibria, which are characterized by the solvability of a system of forward-backward stochastic differential equations and a system of Riccati equations, respectively. The connections and disparities between centralized open-loop and closed-loop Nash equilibria are analyzed. Then, the decentralized control is designed by studying the asymptotic solvability for both open-loop and closed-loop systems. Asymptotically decentralized Nash equilibria are obtained by considering the centralized open-loop and closed-loop Nash equilibria in the infinite-population system, which requires a standard and an asymmetric Riccati equations. The results demonstrate that divergences between the centralized open-loop and closed-loop Nash equilibria in the finite-population system, but the corresponding asymptotically decentralized Nash equilibria in the infinite-population system are consistent. Therefore, the choice of open-loop and closed-loop strategies does not play an essential role in the design of decentralized control for LQ MFGs.
This paper investigates open-loop and feedback solutions of linear quadratic mean field (MF) games with a leader and a large number of followers. The leader first gives its strategy and then all the followers cooperate to optimize the social cost as the sum of their costs. By variational analysis with MF approximations, we obtain a set of open-loop controls of players in terms of solutions to MF forward-backward stochastic differential equations (FBSDEs), which is further shown be to an asymptotic Stackelberg equilibrium. By applying the matrix maximum principle, a set of decentralized feedback strategies is constructed for all the players. For open-loop and feedback solutions, the corresponding optimal costs of all players are explicitly given by virtue of the solutions to two Riccati equations, respectively. The performances of two solutions are compared by the numerical simulation.
This paper studies the spectrum assignment of a class of stochastic systems with multiplicative noise. A novel $α$-spectrum assignment is proposed for discrete-time and continuous-time stochastic systems with multiplicative noise. In particular, $0$-spectrum assignment is equivalent to the pole assignment for the deterministic systems. The main contribution is two-fold: On the one hand, we present the conditions for $α$-spectrum assignment and the design of feedback controllers based on the system parameters. On the other hand, when the system parameters are unknown, we present a stochastic approximation algorithm to learn the feedback gains which guarantee the spectrum of the stochastic systems to achieve the predetermined value. Numerical examples are provided to demonstrate the effectiveness of the proposed algorithms.
The Bundle Adjustment (BA) model is commonly optimized using a nonlinear least squares method, with the Levenberg-Marquardt (L-M) algorithm being a typical choice. However, despite the L-M algorithm's effectiveness, its sensitivity to initial conditions often results in slower convergence when applied to poorly conditioned datasets, motivating the exploration of alternative optimization strategies. This paper introduces a novel algorithm for optimizing the BA model in the context of image sequence alignment for cryo-electron tomography, utilizing optimal control theory to directly optimize general nonlinear functions. The proposed Optimal Control Algorithm (OCA) exhibits superior convergence rates and effectively mitigates the oscillatory behavior frequently observed in L-M algorithm. Extensive experiments on both synthetic and real-world datasets were conducted to evaluate the algorithm's performance. The results demonstrate that the OCA achieves faster convergence compared to the L-M algorithm. Moreover, the incorporation of a bisection-based update procedure significantly enhances the OCA's performance, particularly in poorly initialized datasets. These findings indicate that the OCA can substantially improve the efficiency of 3D reconstructions in cryo-electron tomography.
In the paper, we propose solving optimization problems (OPs) and understanding the Newton method from the optimal control view. We propose a new optimization algorithm based on the optimal control problem (OCP). The algorithm features converging more rapidly than gradient descent, meanwhile, it is superior to Newton's method because it is not divergent in general and can be applied in the case of a singular Hessian matrix. These merits are supported by the convergence analysis for the algorithm in the paper. We also point out that the convergence rate of the proposed algorithm is inversely proportional to the magnitude of the control weight matrix and proportional to the control terminal time inherited from OCP.
This paper proposes several novel optimization algorithms for minimizing a nonlinear objective function. The algorithms are enlightened by the optimal state trajectory of an optimal control problem closely related to the minimized objective function. They are superlinear convergent when appropriate parameters are selected as required. Unlike Newton's method, all of them can be also applied in the case of a singular Hessian matrix. More importantly, by reduction, some of them avoid calculating the inverse of the Hessian matrix or an identical dimension matrix and some of them need only the diagonal elements of the Hessian matrix. In these cases, these algorithms still outperform the gradient descent method. The merits of the proposed optimization algorithm are illustrated by numerical experiments.
This paper focuses on optimal control problem for a class of discrete-time nonlinear systems. In practical applications, computation time is a crucial consideration when solving nonlinear optimal control problems, especially under real-time constraints. While linearization methods are computationally efficient, their inherent low accuracy can compromise control precision and overall performance. To address this challenge, this study proposes a novel approach based on the optimal control method. Firstly, the original optimal control problem is transformed into an equivalent optimization problem, which is resolved using the Pontryagin's maximum principle, and a superlinear convergence algorithm is presented. Furthermore, to improve computation efficiency, explicit formulas for computing both the gradient and hessian matrix of the cost function are proposed. Finally, the effectiveness of the proposed algorithm is validated through simulations and experiments on a linear quadratic regulator problem and an automatic guided vehicle trajectory tracking problem, demonstrating its ability for real-time online precise control.
In this paper, a novel distributed optimization framework has been proposed. The key idea is to convert optimization problems into optimal control problems where the objective of each agent is to design the current control input minimizing the original objective function of itself and updated size for the future time instant. Compared with the existing distributed optimization problem for optimizing a sum of convex objective functions corresponding to multiple agents, we present a distributed optimization algorithm for multi-agents system based on the results from the maximum principle. Moreover, the convergence and superlinear convergence rate are also analyzed stringently.
This paper addresses the optimal control problem of finite-horizon discrete-time nonlinear systems under state and control constraints. A novel numerical algorithm based on optimal control theory is proposed to achieve superior computational efficiency, with the novelty lying in establishing a unified framework that integrates all aspects of algorithm design through the solution of forward and backward difference equations (FBDEs). Firstly, the state and control constraints are transformed using an augmented Lagrangian method (ALM), thereby decomposing the original optimal control problem into several optimization subproblems. These subproblems are then reformulated as new optimal control problem, which are solved through the corresponding FBDEs, resulting in an algorithm with superlinear convergence rate. Furthermore, the gradient and Hessian matrix are computed by iteratively solving FBDEs, thereby accelerating the optimization process. The gradient is obtained through the standard Hamiltonian, while the Hessian matrix is derived by constructing a novel Hamiltonian specifically designed for second-order optimization, transforming each row into an iterative solution of a new set of FBDEs. Finally, the effectiveness of the algorithm is validated through simulation results in automatic guided vehicles (AGV) trajectory tracking control.
In this paper, we propose an online learning-based predictive control (LPC) approach designed for nonlinear systems that lack explicit system dynamics. Unlike traditional model predictive control (MPC) algorithms that rely on known system models to optimize controller outputs, our proposed algorithm integrates a reinforcement learning component to learn optimal policies in real time from the offline dataset and real-time data. Additionally, an optimal control problem (OCP)-based optimization framework is incorporated to enhance real-time computational efficiency while ensuring stability during online operation. Moreover, we rigorously establish the super-linear convergence properties of the algorithm. Finally, extensive simulations are performed to evaluate the feasibility and effectiveness of the proposed approach.
This paper studies the linear-quadratic (LQ) optimal control problem of a class of systems governed by the first-order hyperbolic partial differential equations (PDEs) with final state constraints. The main contribution is to present the solvability condition and the corresponding explicit optimal controller by using the Lagrange multiplier method and the technique of solving forward and backward partial differential equations (FBPDEs). In particular, the result is reduced to the case with zero-valued final state constraints. Several numerical examples are provided to demonstrate the performance of the designed optimal controller.
This paper explores the decentralized control of linear deterministic systems in which different controllers operate based on distinct state information, and extends the findings to the output feedback scenario. Assuming the controllers have a linear state feedback structure, we derive the expression for the controller gain matrices using the matrix maximum principle. This results in an implicit expression that couples the gain matrices with the state. By reformulating the backward Riccati equation as a forward equation, we overcome the coupling between the backward Riccati equation and the forward state equation. Additionally, we employ a gradient descent algorithm to find the solution to the implicit equation. This approach is validated through simulation examples.