Time-varying Optimization Problem Research Articles

Distributed Time-Varying Optimization-An Output Regulation Approach.

This article deals with distributed algorithm design for time-varying optimization problems, which include unconstrained time-varying optimization and a special constrained problem commonly known as a resource allocation problem. The time-varying nature exists in the individual cost functions and the demand functions, and they are then captured by neutrally stable linear dynamic systems known as exosystems. To address the time-varying nature, new distributed algorithm structures are developed and two algorithms are designed for distributed time-varying optimization (DTVO) and distributed time-varying optimal resource allocation (DTVORA) to ensure that there exist time-varying solutions and the solution states will converge to the time-varying solutions. The driving terms for tracking the variation in the solutions are designed using exosystem dynamics. Rigorous convergence analyses are carried out using Lyapunov theory, and the examples are included to demonstrate the potential applications of the two proposed algorithms.

Maximum Load Consumption Capacity Maintenance of Distributed Storage Devices Based on Time-Varying Neurodynamic Algorithm

A charge and discharge management scheme is proposed. The stored electric energy in distributed storage devices will converge to the consistent. The consistency of the stored electric energy of the devices helps to maintain the maximum load capacity and maximum consumption capacity of distributed storage devices. The charging and discharging process is constructed as a time-varying optimization problem, and the proposed algorithm can respond to the time-varying parameters of the distributed storage devices in real time. The time-varying neurodynamic algorithms can obtain time-varying optimal solution trajectories to give the optimal charging and discharging strategy in real time. In addition, the proposed approach in this paper focuses on the privacy protection of device data. Each device can calculate the power of discharging or charging by communicating with the partially connected nodes. Numerical simulations of the proposed scheme in the paper are given to verify the effectiveness of the scheme. Numerical simulations show that our scheme can make the electric energy stored in each storage device converge and maintain the maximum load capacity or maximum consumption capacity of the whole distributed storage device.

Distributed time-varying optimization control protocol for multi-agent systems via finite-time consensus approach

This paper addresses a distributed time-varying optimization problem with inequality constraints based on multi-agent systems over switching communication graphs. To reduce the influence of time-varying inequality constraints, an exact penalty method and smoothing technique are employed. Then, a Hessian-based distributed control protocol is presented to seek the time-varying optimal solution of the distributed time-varying optimization problem by virtue of only local information and interaction. It is shown that all agents not only achieve finite-time consensus but also track the time-varying global optimal target eventually. Compared with the existing distributed optimization protocols, the proposed control protocol is suitable for more general distributed time-varying optimization problems and enjoys high-efficiency convergence. Finally, numerical examples and experiment on moving target tracking of Unmanned Aircraft Vehicle (UAV) are performed to illustrate the effectiveness of the proposed control protocol.

An inertial neurodynamic algorithm for collaborative time-varying energy management for energy internet containing distributed energy resources

This paper investigates the energy management of an Energy Internet (EI) that integrates multiple energy networks. Based on the structure of EI and the fluctuation of load, an optimal energy management model considering part of the load as time-varying factor is proposed. Under the influence of load, the actual energy management problem (EMP) is formulated as a time-varying optimization problem subject to a set of global and local constraints. Its optimal solution varies continuously with time, which leads to difficulties in obtaining optimal solutions by conventional algorithms. Consequently, to obtain better tracking performance, this paper proposes a distributed inertial projection neurodynamics algorithm (DIPNA) based on the Nesterov accelerated gradient descent method. This algorithm tracks the optimal solution of the EMP with a fast convergence rate O1/t2, providing a referenceable active power for each energy unit in real time. Finally, the performance evaluation results demonstrated the effectiveness and fast convergence of the proposed algorithm.

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.

Convex risk-bounded continuous-time trajectory planning and tube design in uncertain nonconvex environments

In this paper, we address the trajectory planning problem in uncertain nonconvex static and dynamic environments that contain obstacles with probabilistic location, size, and geometry. To address this problem, we provide a risk-bounded trajectory planning method that looks for continuous-time trajectories with guaranteed bounded risk over the planning time horizon. Risk is defined as the probability of collision with uncertain obstacles. Existing approaches to address risk-bounded trajectory planning problems either are limited to Gaussian uncertainties and convex obstacles or rely on sampling-based methods that need uncertainty samples and time discretization. To address the risk-bounded trajectory planning problem, we leverage the notion of risk contours to transform the risk-bounded planning problem into a deterministic optimization problem. Risk contours are the set of all points in the uncertain environment with guaranteed bounded risk. The obtained deterministic optimization is, in general, nonlinear and nonconvex time-varying optimization. We provide convex methods based on sum-of-squares optimization to efficiently solve the obtained nonconvex time-varying optimization problem and obtain the continuous-time risk-bounded trajectories without time discretization. The provided approach deals with arbitrary (and known) probabilistic uncertainties, nonconvex and nonlinear, static and dynamic obstacles, and is suitable for online trajectory planning problems. In addition, we provide convex methods based on sum-of-squares optimization to build the max-sized tube with respect to its parameterization along the trajectory so that any state inside the tube is guaranteed to have bounded risk.

Distributed online prediction optimization algorithm for distributed energy resources considering the multi-periods optimal operation

In this paper, a distributed online prediction optimization algorithm is proposed to optimize the operation of distributed energy resources (DERs) considering multi-period constraints. First, we build a time-varying operation optimization model of DERs in multi-area distribution networks considering the operation constraints among different periods (e.g., the relationship of DERs energy within a certain amount of time). Second, we design an online prediction algorithm to solve the time-varying model in a distributed way. Specifically, it decouples the sensitivities of power flow states among different distribution areas. Based on this, a model is set up to characterize the mapping relationship among power flow states and the increments of DER output powers among different periods, and to predict the power flow states in the future only depending on the local and some aggregated information of each area. Thus, the solution of the time-varying optimization problem is decomposed into solving several subproblems in a distributed way. Finally, the proposed algorithm is verified in a 502-node distribution system. It achieves the optimal operation of DERs with satisfying the constraints of the energy plan formulated to reserve enough energy for the regulation in future periods. Compared with the centralized method, the proposed algorithm achieves similar optimization results with significant reduction of calculation time.

Time-Varying Optimization of Networked Systems With Human Preferences

This article considers a time-varying optimization problem associated with a network of systems, with each of the systems shared by (and affecting) a number of individuals. The objective is to minimize cost functions associated with the individuals' preferences, which are unknown, subject to time-varying constraints that capture physical or operational limits of the network. To this end, this article develops a distributed online optimization algorithm with concurrent learning of the cost functions. The cost functions are learned on-the-fly based on the users' feedback (provided at irregular intervals) by leveraging tools from the shape-constrained Gaussian process. The online algorithm is based on a primal–dual method and acts effectively in a closed-loop fashion where: First, users' feedback is utilized to estimate the cost, and second, measurements from the network are utilized in the algorithmic steps to bypass the need for sensing of (unknown) exogenous inputs of the network. The performance of the algorithm is analyzed in terms of dynamic network regret and constraint violation. Numerical examples are presented in the context of real-time optimization of distributed energy resources.

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.

On the Absence of Spurious Local Trajectories in Time-Varying Nonconvex Optimization

In this article, we study the landscape of an online nonconvex optimization problem, for which the input data vary over time and the solution is a trajectory rather than a single point. To understand the complexity of finding a global solution of this problem, we introduce the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">spurious (i.e., nonglobal) local trajectory</i> as a generalization to the notion of spurious local solution in nonconvex (time-invariant) optimization. We develop an ordinary differential equation (ODE) associated with a time-varying nonlinear dynamical system which, at limit, characterizes the spurious local solutions of the time-varying optimization problem. We prove that the absence of spurious local trajectory is closely related to the transient behavior of the developed system. In particular, we show that if the problem is time-varying, the data variation may force all of the ODE trajectories initialized at arbitrary local minima at the initial time to gradually converge to the global solution trajectory. We study the Jacobian of the dynamical system along a local minimum trajectory and show how its eigenvalues are manipulated by the natural data variation in the problem, which may consequently trigger escaping poor local minima over time.

Incorporate day-ahead robustness and real-time incentives for electricity market design

In this paper, we propose a two-stage electricity market framework to explore the participation of distributed energy resources (DERs) in a day-ahead (DA) market and a real-time (RT) market. The objective is to determine the optimal bidding strategies of the aggregated DERs in the DA market and generate online incentive signals for DER-owners to optimize the social-welfare taking into account network operational constraints. Distributionally robust optimization is used to explicitly incorporate data-based statistical information of renewable forecasts into the supply/demand decisions in the DA market. We evaluate the conservativeness of bidding strategies distinguished by different risk aversion settings. In the RT market, a bi-level time-varying optimization problem is proposed to design the online incentive signals to tradeoff the RT imbalance penalty for distribution system operators (DSOs) and the costs of individual DER-owners. This enables tracking their optimal dispatch to provide fast balancing services, in the presence of time-varying network states while satisfying the voltage regulation requirement. Simulation results on both DA wholesale market and RT balancing market demonstrate the necessity of this two-stage design, and its robustness to uncertainties, the performance of convergence, the tracking ability and the feasibility of the resulting network operations.

Distributed Time-Varying Optimization of Second-Order Multiagent Systems Under Limited Interaction Ranges.

This article investigates the distributed time-varying optimization problem for second-order multiagent systems (MASs) under limited interaction ranges. The goal is to seek the minimum of the sum of local time-varying cost functions (CFs), where each CF is only available to the corresponding agent. Limited communication range refers to the scenario where the agents have limited sensing and communication capabilities, that is, a pair of agents can communicate with each other only if their distance is within a certain range. To handle such a problem, a new continuous connectivity-preserving mechanism is presented to preserve the connectivity of the considered network. Then, two distributed optimization algorithms are presented to solve the optimization problem with time-varying CFs and time-invariant CFs, respectively. Theoretical analysis and two numerical examples are provided to verify the effectiveness of the methods.

Computational Neural Dynamics Model for Time-Variant Constrained Nonlinear Optimization Applied to Winner-Take-All Operation

Although time-variant nonlinear optimization with equality and inequality constraints (TVNOEIC) is a widespread problem in real life, the research on this issue is much less than the time-invariant one. Large lagging errors may not be inevitable if the traditional models designed for time-invariant optimizations are exploited to time-variant ones. To eliminate the lagging errors, models for handling time-variant optimization problems are urgently required. Recently, neural networks and the related neural dynamics approaches have witnessed rapid developments and made significant contributions to the online solution of various problems. On this basis, a noise-suppressing discrete-time neural dynamics (NSDTND) model is proposed in this article for solving the TVNOEIC problem. Theoretical analysis is presented to prove that the residual error of the proposed model is tiny enough to estimate the actual solution even in the presence of noise perturbation. In addition, numerical simulations, including an application on winner-take-all operation, are provided to further demonstrate the effectiveness and robustness of the proposed NSDTND model.

Time-Varying Optimization-Based Approach for Distributed Formation of Uncertain Euler-Lagrange Systems.

We investigate a distributed time-varying formation control problem for an uncertain Euler-Lagrange system. A time-varying optimization-based approach is proposed. Based on this approach, the robots can achieve the expected formation configuration and meanwhile optimize a global objective function using only neighboring and local information. We consider the time-varying optimization where the objective functions can change in real time. In this case, the consensus-based formation tracking control issues and formation containment tracking control issues in the literature can be solved by the proposed approach. By a penalty-based method, the robots' states asymptotically converge to the estimated optimal solution to an equivalent time-varying optimization problem, whose optimal solution can achieve simultaneous formation and optimization. Furthermore, we consider two more general scenarios: 1) the local objective functions can have non-neighbor's information and 2) the optimization problems can have inequality constraints.

Quantitative Sensitivity Bounds for Nonlinear Programming and Time-Varying Optimization

Inspired by classical sensitivity results for nonlinear optimization, we derive and discuss new quantitative bounds to characterize the solution map and dual variables of a parametrized nonlinear program. In particular, we derive explicit expressions for the local and global Lipschitz constants of the solution map of nonconvex or convex optimization problems, respectively. Our results are geared towards the study of time-varying optimization problems, which are commonplace in various applications of online optimization, including power systems, robotics, signal processing, and more. In this context, our results can be used to bound the rate of change of the optimizer. To illustrate the use of our sensitivity bounds we generalize existing arguments to quantify the tracking performance of continuous-time, monotone running algorithms. Furthermore, we introduce a new continuous-time running algorithm for time-varying constrained optimization, which we model as a so-called perturbed sweeping process. For this discontinuous scheme we establish an explicit bound on the asymptotic solution tracking for a class of convex problems.

Streaming Solutions for Time-Varying Optimization Problems

This paper studies streaming optimization problems that have objectives of the form <inline-formula><tex-math notation="LaTeX">$ \sum _{t=1}^{T}f(x_{t-1},x_{t})$</tex-math></inline-formula>. In particular, we are interested in how the solution <inline-formula><tex-math notation="LaTeX">$\hat{x}_{t|T}$</tex-math></inline-formula> for the <inline-formula><tex-math notation="LaTeX">$t$</tex-math></inline-formula>th frame of variables changes as <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula> increases. While incrementing <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula> and adding a new functional and a new set of variables does in general change the solution everywhere, we give conditions under which <inline-formula><tex-math notation="LaTeX">$\hat{x}_{t|T}$</tex-math></inline-formula> converges to a limit point <inline-formula><tex-math notation="LaTeX">$x^*_{t}$</tex-math></inline-formula> at a linear rate as <inline-formula><tex-math notation="LaTeX">$T\rightarrow \infty$</tex-math></inline-formula>. As a consequence, we are able to derive theoretical guarantees for algorithms with limited memory, showing that limiting the solution updates to only a small number of frames in the past sacrifices almost nothing in accuracy. We also present a new efficient Newton online algorithm (NOA), inspired by these results, that updates the solution with fixed per-iteration complexity of <inline-formula><tex-math notation="LaTeX">$ \mathcal{O}(3Bn^{3})$</tex-math></inline-formula>, independent of <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$B$</tex-math></inline-formula> corresponds to how far in the past the variables are updated, and <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> is the size of a single block-vector. Two streaming optimization examples, online reconstruction from non-uniform samples and inhomogeneous Poisson intensity estimation, support the theoretical results and show how the algorithm can be used in practice.

Dynamical behavior and optimal control of a vector-borne diseases model on bipartite networks

For better application of vaccination control and quarantine control, a novel vector-borne diseases model with two control strategies on bipartite networks is proposed. By using complex network theory and Lyapunov function method, the basic reproduction number is obtained, and the global stability of disease-free equilibrium and endemic equilibrium are proved. Then the time-varying optimal control problem for reducing control cost is investigated. Using the optimal control theory, the existence and uniqueness of the optimal control and the model’s optimal solution are both studied. Finally, numerical simulations are carried out to verify the analysis results and the efficiency of the optimal control.

Bounds for the Tracking Error of First-Order Online Optimization Methods

This paper investigates online algorithms for smooth time-varying optimization problems, focusing first on methods with constant step-size, momentum, and extrapolation-length. Assuming strong convexity, precise results for the tracking iterate error (the limit supremum of the norm of the difference between the optimal solution and the iterates) for online gradient descent are derived. The paper then considers a general first-order framework, where a universal lower bound on the tracking iterate error is established. Furthermore, a method using “long-steps” is proposed and shown to achieve the lower bound up to a fixed constant. This method is then compared with online gradient descent for specific examples. Finally, the paper analyzes the effect of regularization when the cost is not strongly convex. With regularization, it is possible to achieve a non-regret bound. The paper ends by testing the accelerated and regularized methods on synthetic time-varying least-squares and logistic regression problems, respectively.

An Online Framework for Joint Network Selection and Service Placement in Mobile Edge Computing

With the rapid development and deployment of 5G wireless technology, mobile edge computing (MEC) has emerged as a new computing paradigm to facilitate a large variety of infrastructures at the network edge to reduce user-perceived communication delay. One of the fundamental problems in this new paradigm is to preserve satisfactory quality-of-service (QoS) for mobile users in light of densely dispersed wireless communication environment and often capacity-constrained MEC nodes. Such user-perceived QoS, typically in terms of the end-to-end delay, is highly vulnerable to both access network bottleneck and communication delay. Previous works have primarily focused on optimizing the communication delay through dynamic service placement, while ignoring the critical effect of access network selection on the access delay. In this work, we study the problem of jointly optimizing the access network selection and service placement for MEC, with the objective of improving the QoS in a cost-efficient manner by judiciously balancing the access delay, communication delay, and service switching cost. Specifically, we propose an efficient online framework to decompose a long-term time-varying optimization problem into a series of one-shot subproblems. To address the NP-hardness of the one-shot problem, we design a computationally-efficient two-phase algorithm based on matching and game theory, which achieves a near-optimal solution. Both rigorous theoretical analysis on the optimality gap and extensive trace-driven simulations are conducted to validate the efficacy of our proposed solution.

Distributed Time-Varying Optimization With State-Dependent Gains: Algorithms and Experiments

This brief addresses distributed continuous-time optimization problems with time-varying objective functions. The goal is for multiple agents to cooperatively minimize the sum of local time-varying objective functions with only local interaction and information while explicitly taking into account distributed adaptive gain design. Here, the optimal point is time varying and creates an optimal trajectory. First, for the unconstrained case, a distributed nonsmooth algorithm coupled with a state-dependent gain is proposed. It is shown that the interaction gain for each agent can be computed according to the variation of the Hessian and gradient information of the convex local objective functions so that the algorithm can solve the time-varying optimization problem without imposing a bound on any information about the local objective functions. Second, for the case where there exist common time-varying linear equality constraints, an extended algorithm is presented, where local Lagrangian functions are introduced to address the equality constraints. The asymptotic convergence of both algorithms to the optimal solution is proved. Numerical simulations are presented to illustrate the theoretical results. In addition, the one proposed algorithm is experimentally implemented and validated on a multi-Crazyflie platform.