Time-varying Convex Optimization Research Articles

General Lyapunov Stability and its Application to Time-Varying Convex Optimization

General Lyapunov Stability and its Application to Time-Varying Convex Optimization

Sign Hessian-Weighted Gradient Algorithms for Distributed Time-Varying Convex Optimization

Sign Hessian-Weighted Gradient Algorithms for Distributed Time-Varying Convex Optimization

Distributed Newton method for time-varying convex optimization with backward Euler prediction

<p>We investigated the challenge of unconstrained distributed optimization with a time-varying objective function, employing a prediction-correction approach. Our method introduced a backward Euler prediction step that used the differential information from consecutive moments to forecast the trajectory's future direction. This predicted value was then refined through an iterative correction process. Our analysis and experimental results demonstrated that this approach effectively addresses the optimization problem without requiring the computation of the Hessian matrix's inverse.</p>

A Hessian-based zeroing neurodynamic approach for quaternion-variable time-varying constrained optimization problems

In this paper, a quaternion-variable zeroing neurodynamic (ZND) is designed, in order to address a class of quaternion-variable time-varying convex optimization problems (TCOP) with equality constraints and general inequality constraints. Through utilizing the Karush–Kuhn–Tucker (KKT) condition, the TCOP is transformed into a time-varying equation system for solving. During this process, the introduction of two approaches distinguishes our works in solving TCOPs. On the one hand, the penalty approach is utilized to reduce the influence of time-varying inequality constraints, and thus, makes the ZND more suitable for TCOPs with general and complex inequality constraints. On the other hand, the approach to simplify the KKT condition significantly simplifies the structure of the proposed ZND by reducing the neurons’ number. The proposed ZND realizes efficient convergence while keeping the structure as simple as possible, making it easy to apply in engineering tasks. Besides, to improve its efficiency further, a nonlinear activated function is utilized to achieve a faster convergent rate, by which the convergent rate of the ZND is accelerated to finite-time convergence. Apart from that, the ZND here also owes superior robustness. To the best of our knowledge, this is the first time to extend the ZND for solving TCOP to the quaternion domain. Finally, numerical examples and a robot manipulator control experiment further explain the effectiveness of the proposed ZND.

Time-varying distributed optimization problem with inequality constraints

This paper discusses a distributed time-varying convex optimization problem that agents have different Hessian matrices with inequality constraints. The objective is to minimize the sum of local time-varying objective functions of agents constrained by time-varying inequalities. Under the condition of undirected connected graph, a distributed continuous time consistency algorithm is designed based on average consensus estimator, sign function and log-barrier penalty function. The main idea of the algorithm proposed in this paper is to use the estimator to estimate the global information, make the state of agents reach consensus and achieve gradient descent to track the optimal solution. Theoretical findings show that all agents can reach an agreement, the proposed algorithm can track the optimal solution of the time-varying optimization problem. The effectiveness of the theoretical results is verified through a numerical examples.

Extrapolation-Based Prediction-Correction Methods for Time-varying Convex Optimization

In this paper, we focus on the solution of online optimization problems that arise often in signal processing and machine learning, in which we have access to streaming sources of data. We discuss algorithms for online optimization based on the prediction-correction paradigm, both in the primal and dual space. In particular, we leverage the typical regularized least-squares structure appearing in many signal processing problems to propose a novel and tailored prediction strategy, which we call extrapolation-based. By using tools from operator theory, we then analyze the convergence of the proposed methods as applied both to primal and dual problems, deriving an explicit bound for the tracking error, that is, the distance from the time-varying optimal solution. We further discuss the empirical performance of the algorithm when applied to signal processing, machine learning, and robotics problems.

Distributed Time-Varying Convex Optimization With Dynamic Quantization.

In this work, we design a distributed algorithm for time-varying convex optimization over networks with quantized communications. Each agent has its local time-varying objective function, while the agents need to cooperatively track the optimal solution trajectories of global time-varying functions. The distributed algorithm is motivated by the alternating direction method of multipliers, but the agents can only share quantization information through an undirected graph. To reduce the tracking error due to information loss in quantization, we apply the dynamic quantization scheme with a decaying scaling function. The tracking error is explicitly characterized with respect to the limit of the decaying scaling function in quantization. Furthermore, we are able to show that the algorithm could asymptotically track the optimal solution when time-varying functions converge, even with quantization information loss. Finally, the theoretical results are validated via numerical simulation.

Fixed-Time Algorithms for Time-Varying Convex Optimization

To resolve the time-varying convex optimization problems with the cost function, the constraints or both being time dependent, in this brief we investigate a novel type of fixed-time algorithms. First, with the unconstrained time-varying optimization problem considered, a general framework algorithm is developed for tracking its optimal trajectory within fixed time, which contains the gradient flow-based scheme and Newton-type method as its special cases. Then, considering the equality constraint being involved in the time-varying optimization problem, we design another algorithm with fixed-time convergence, which includes Newton-type scheme as its special case. To verify that the given approach achieves fixed-time convergence, the simulation result is given with first-order Euler discretization.

A Totally Asynchronous Algorithm for Time-Varying Convex Optimization Problems

This paper presents a decentralized algorithm for a team of agents to track time-varying fixed points that are the solutions to time-varying convex optimization problems. The algorithm is first-order, and it allows for total asynchrony, i.e., all communications and computations can occur with arbitrary timing and arbitrary (finite) delays. Convergence rates are computed in terms of the communications and computations that agents execute, without specifying when they must occur. These rates apply to convergence to the minimum of each individual objective function, as well as agents’ long-run behavior as their objective functions change. Then, to improve the usage of limited communication and computation resources, we optimize the timing of agents’ operations relative to changes in their objective functions to minimize fixed point tracking error over time. Simulation results illustrate these developments.

Online optimization of switched LTI systems using continuous-time and hybrid accelerated gradient flows

This paper studies the design of feedback controllers to steer a switching linear dynamical system to the solution trajectory of a time-varying convex optimization problem. We propose two types of controllers: (i) a continuous controller inspired by the online gradient descent method, and (ii) a hybrid controller that can be interpreted as an online version of Nesterov’s accelerated gradient method with restarts of the state variables. By design, the controllers continuously steer the system toward a time-varying optimal equilibrium point without requiring knowledge of exogenous disturbances affecting the system. For cost functions that are smooth and satisfy the Polyak–Łojasiewicz inequality, we demonstrate that the online gradient-flow controller ensures uniform global exponential stability when the time scales of the system and controller are sufficiently separated and the switching signal of the system varies slowly on average. For cost functions that are strongly convex, we show that the hybrid accelerated controller can outperform the continuous gradient descent method. When the cost function is not strongly convex, we show that the hybrid accelerated method guarantees global practical asymptotic stability.

Online Learning Over Dynamic Graphs via Distributed Proximal Gradient Algorithm

We consider the problem of tracking the minimum of a time-varying convex optimization problem over a dynamic graph. Motivated by target tracking and parameter estimation problems in intermittently connected robotic and sensor networks, the goal is to design a distributed algorithm capable of handling nondifferentiable regularization penalties. The proposed proximal online gradient descent algorithm is built to run in a fully decentralized manner and utilizes consensus updates over possibly disconnected graphs. The performance of the proposed algorithm is analyzed by developing bounds on its dynamic regret in terms of the cumulative path length of the time-varying optimum. It is shown that as compared to the centralized case, the dynamic regret incurred by the proposed algorithm over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> time slots is worse by a factor of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\log (T)$</tex-math></inline-formula> only, despite the disconnected and time-varying network topology. The empirical performance of the proposed algorithm is tested on the distributed dynamic sparse recovery problem, where it is shown to incur a dynamic regret that is close to that of the centralized algorithm.

A continuous-time consensus algorithm using neurodynamic system for distributed time-varying optimization with inequality constraints

In this paper, a distributed time-varying convex optimization problem with inequality constraints is discussed based on neurodynamic system. The goal is to minimize the sum of agents’ local time-varying objective functions subject to some time-varying inequality constraints, each of which is known only to an individual agent. Here, the optimal solution is time-varying instead of constant. Under an undirected and connected graph, a distributed continuous-time consensus algorithm is designed by using neurodynamic system, signum functions and log-barrier penalty functions. The proposed algorithm can be understood through two parts: one part is used to reach consensus and the other is used to achieve gradient descent to track the optimal solution. Theoretical studies indicate that all agents will achieve consensus and the proposed algorithm can track the optimal solution of the time-varying convex problem. Two numerical examples are provided to validate the theoretical results.

Online discriminative graph learning from multi-class smooth signals

Graph signal processing (GSP) is a key tool for satisfying the growing demand for information processing over networks. However, the success of GSP in downstream learning and inference tasks such as supervised classification is heavily dependent on the prior identification of the relational structures. Graphs are natural descriptors of the relationships between entities of complex environments. The underlying graph is not readily detectable in many cases and one has to infer the topology from the observed signals which admit certain regularity over the sought representation. Assuming that signals are smooth over the latent class-specific graph is the notion we build upon. Firstly, we address the problem of graph signal classification by proposing a novel framework for discriminative graph learning. To learn discriminative graphs, we invoke the assumption that signals belonging to each class are smooth with respect to the corresponding graph while maintaining non-smoothness with respect to the graphs corresponding to other classes. Discriminative features are extracted via graph Fourier transform (GFT) of the learned representations and used in downstream learning tasks. Secondly, we extend our work to tackle increasingly dynamic environments and real-time topology inference. To this end, we adopt recent advances of GSP and time-varying convex optimization. We develop a proximal gradient (PG) method which can be adapted to situations where the data are acquired on-the-fly. Beyond discrimination, this is the first work that addresses the problem of dynamic graph learning from smooth signals where the sought network alters slowly. The introduced online framework is guaranteed to track the optimal time-varying batch solution under mild technical conditions. The validation of the proposed frameworks is comprehensively investigated using both synthetic and real data. The proposed classification pipeline outperforms the-state-of-the-art methods when applied to the problem of emotion classification based on electroencephalogram (EEG) data. We also perform network-based analysis of epileptic seizures using electrocorticography (ECoG) records. Moreover, by applying our method to financial data, our approach infers the relationships between the stock-price behavior of leading US companies and the recent events including the COVID-19 pandemic.

Time-Varying Convex Optimization: Time-Structured Algorithms and Applications

Optimization underpins many of the challenges that science and technology face on a daily basis. Recent years have witnessed a major shift from traditional optimization paradigms grounded on batch algorithms for medium-scale problems to challenging dynamic, time-varying, and even huge-size settings. This is driven by technological transformations that converted infrastructural and social platforms into complex and dynamic networked systems with even pervasive sensing and computing capabilities. This article reviews a broad class of state-of-the-art algorithms for time-varying optimization, with an eye to performing both algorithmic development and performance analysis. It offers a comprehensive overview of available tools and methods and unveils open challenges in application domains of broad range of interest. The real-world examples presented include smart power systems, robotics, machine learning, and data analytics, highlighting domain-specific issues and solutions. The ultimate goal is to exemplify wide engineering relevance of analytical tools and pertinent theoretical foundations.

An inverse-free Zhang neural dynamic for time-varying convex optimization problems with equality and affine inequality constraints

Time-varying convex optimization problems have attracted a great deal of attention in many fields due to its widespread application. Particularly, the approach to time-varying convex optimization problems with equality and affine inequality constraints simultaneously is a comprehensive but complicated problem at present. In this paper, three types of inverse-free Zhang neural dynamic (ZND) including two noise-tolerance ZND models are proposed and investigated for solving time-varying convex optimization problems with equality and affine inequality constraints. It is noted that the proposed noise-tolerance ZND models own the ability to suppress noise and one of those even achieves finite-time convergency. Compared with previous work, the proposed inverse-free ZND models effectively reduce the computation complexity by simplifying the model structure and conquer the problem of solving real-time matrix inverse during the computation process, which is more appropriate for a wider practical application.

Online Optimization as a Feedback Controller: Stability and Tracking

This paper develops and analyzes feedback-based online optimization methods to regulate the output of a linear time invariant (LTI) dynamical system to the optimal solution of a time-varying convex optimization problem. The design of the algorithm is based on continuous-time primal-dual dynamics, properly modified to incorporate feedback from the LTI dynamical system, applied to a proximal augmented Lagrangian function. The resultant closed-loop algorithm tracks the solution of the time-varying optimization problem without requiring knowledge of (time varying) disturbances in the dynamical system. The analysis leverages integral quadratic constraints to provide linear matrix inequality (LMI) conditions that guarantee global exponential stability and bounded tracking error. Analytical results show that under a sufficient time-scale separation between the dynamics of the LTI dynamical system and the algorithm, the LMI conditions can be always satisfied. This paper further proposes a modified algorithm that can track an approximate solution trajectory of the constrained optimization problem under less restrictive assumptions. As an illustrative example, the proposed algorithms are showcased for power transmission systems, to compress the time scales between secondary and tertiary control, and allow to simultaneously power rebalancing and tracking of the DC optimal power flow points.

Distributed Time-Varying Convex Optimization for a Class of Nonlinear Multiagent Systems

This paper studies the distributed time-varying convex optimization problem for a class of nonlinear multiagent systems. An auxiliary system is first designed to estimate the global optimal state and its high-order derivatives by means of a distributed algorithm. Relying on this auxiliary system, the solution of the distributed time-varying convex optimization problem is converted into that of a tracking problem for a class of nonlinear systems. A tracking control scheme is then proposed based on the back-stepping strategy. Simulations are provided to validate the theoretical results.

Online Primal-Dual Methods With Measurement Feedback for Time-Varying Convex Optimization

This paper addresses the design and analysis of feedback-based online algorithms to control systems or networked systems based on performance objectives and engineering constraints that may evolve over time. The emerging time-varying convex optimization formalism is leveraged to model optimal operational trajectories of the systems, as well as explicit local and network-level operational constraints. Departing from existing batch and feed-forward optimization approaches, the design of the algorithms capitalizes on an online implementation of primal-dual projected-gradient methods; the gradient steps are, however, suitably modified to accommodate feedback from the system in the form of measurements, hence, the term “online optimization with feedback.” By virtue of this approach, the resultant algorithms can cope with model mismatches in the algebraic representation of the system states and outputs, they avoid pervasive measurements of exogenous inputs, and they naturally lend themselves to a distributed implementation. Under suitable assumptions, analytical convergence claims are established in terms of dynamic regret. Furthermore, when the synthesis of the feedback-based online algorithms is based on a regularized Lagrangian function, Q-linear convergence to solutions of the time-varying optimization problem is shown.

Decentralized Prediction-Correction Methods for Networked Time-Varying Convex Optimization

We develop algorithms that find and track the optimal solution trajectory of time-varying convex optimization problems that consist of local and network-related objectives. The algorithms are derived from the prediction-correction methodology, which corresponds to a strategy where the time-varying problem is sampled at discrete time instances, and then, a sequence is generated via alternatively executing predictions on how the optimizers at the next time sample are changing and corrections on how they actually have changed. Prediction is based on how the optimality conditions evolve in time, while correction is based on a gradient or Newton method, leading to decentralized prediction-correction gradient and decentralized prediction-correction Newton. We extend these methods to cases where the knowledge on how the optimization programs are changing in time is only approximate and propose decentralized approximate prediction-correction gradient and decentralized approximate prediction-correction Newton. Convergence properties of all the proposed methods are studied and empirical performance is shown on an application of a resource allocation problem in a wireless network. We observe that the proposed methods outperform existing running algorithms by orders of magnitude. The numerical results showcase a tradeoff between convergence accuracy, sampling period, and network communications.

Event-Based Networked Islanding Detection for Distributed Solar PV Generation Systems

This paper proposes an event-based networked set-membership filtering method to detect islanding fault for distributed grid-connected solar photovoltaic generation systems. The method enables each set-membership filter to offer an ellipsoidal estimation set, which is used to judge whether or not the islanding fault happens. When islanding fault happens, the intersection of the ellipsoids is empty, and when islanding fault is free, the intersection of the ellipsoids is nonempty. In the filtering scheme, a novel event-triggered mechanism is proposed to reduce the transmission frequency for saving the communication resources. The condition of the existence of the set-membership algorithm is derived by a time-varying convex optimization approach. A simulation experiment and a comparative experiment are provided using Sim-Power-Systems implementation based on a 2-kW single-phase grid-connected power generation system to illustrate the effectiveness of the proposed method for the detection of the islanding fault and the reduction of the resource consumption, respectively.