Consensus-based Optimization Algorithm Research Articles

Time-discrete momentum consensus-based optimization algorithm and its application to Lyapunov function approximation

In this paper, we study a discrete momentum consensus-based optimization (Momentum-CBO) algorithm which corresponds to a second-order generalization of the discrete first-order CBO [S.-Y. Ha, S. Jin and D. Kim, Convergence of a first-order consensus-based global optimization algorithm, Math. Models Methods Appl. Sci. 30 (2020) 2417–2444]. The proposed algorithm can be understood as the modification of ADAM-CBO, replacing the normalization term by unity. For the proposed Momentum-CBO, we provide a sufficient framework which guarantees the convergence of algorithm toward a global minimum of the objective function. Moreover, we present several experimental results showing that Momentum-CBO has an improved success rate of finding the global minimum compared to vanilla-CBO and show the stability of Momentum-CBO under different initialization schemes. We also show that Momentum-CBO can be used as the alternative of ADAM-CBO which does not have a proper convergence analysis. Finally, we give an application of Momentum-CBO for Lyapunov function approximation using symbolic regression techniques.

Convergence analysis of the discrete consensus-based optimization algorithm with random batch interactions and heterogeneous noises

We present stochastic consensus and convergence of the discrete consensus-based optimization (CBO) algorithm with random batch interactions and heterogeneous external noises. Despite the wide applications and successful performance in many practical simulations, the convergence of the discrete CBO algorithm was not rigorously investigated in such a generality. In this work, we introduce a generalized discrete CBO algorithm with a weighted representative point and random batch interactions, and show that the proposed discrete CBO algorithm exhibits stochastic consensus and convergence toward the common equilibrium state exponentially fast under suitable assumptions on system parameters. For this, we recast the given CBO algorithm with random batch interactions as a discrete consensus model with a random switching network topology, and then we use the mixing property of interactions over sufficiently long time interval to derive stochastic consensus and convergence estimates in mean square and almost sure senses. Our proposed analysis significantly improves earlier works on the convergence analysis of CBO models with full batch interactions and homogeneous external noises.

Stochastic consensus dynamics for nonconvex optimization on the Stiefel manifold: Mean-field limit and convergence

We study a consensus-based method for minimizing a nonconvex function over the Stiefel manifold. The consensus dynamics consists of stochastic differential equations for interacting particle system, whose trajectory is guaranteed to stay on the Stiefel manifold. For the proposed model, we prove the mean-field limit of the stochastic system toward a nonlinear Fokker–Planck equation on the Stiefel manifold. Moreover, we provide a sufficient condition on the parameter and the initial data, so that the solution to the Fokker–Planck equation is asymptotically concentrated on the point near a global optimizer. To implement our consensus-based optimization (CBO) algorithm, we provide two algorithms; one is improved from the algorithm suggested in our previous work, and the other is based on an entirely different approach, namely the Cayley transformation. We validate the CBO algorithms on the various test problems on the Stiefel manifold.

Consensus-Based Optimization for 3D Human Pose Estimation in Camera Coordinates

3D human pose estimation is frequently seen as the task of estimating 3D poses relative to the root body joint. Alternatively, we propose a 3D human pose estimation method in camera coordinates, which allows effective combination of 2D annotated data and 3D poses and a straightforward multi-view generalization. To that end, we cast the problem as a view frustum space pose estimation, where absolute depth prediction and joint relative depth estimations are disentangled. Final 3D predictions are obtained in camera coordinates by the inverse camera projection. Based on this, we also present a consensus-based optimization algorithm for multi-view predictions from uncalibrated images, which requires a single monocular training procedure. Although our method is indirectly tied to the training camera intrinsics, it still converges for cameras with different intrinsic parameters, resulting in coherent estimations up to a scale factor. Our method improves the state of the art on well known 3D human pose datasets, reducing the prediction error by 32% in the most common benchmark. We also reported our results in absolute pose position error, achieving 80 mm for monocular estimations and 51 mm for multi-view, on average. Source code is available at https://github.com/dluvizon/3d-pose-consensus .

Decentralized and Model-Free Federated Learning: Consensus-Based Distillation in Function Space

This paper proposes a fully decentralized federated learning (FL) scheme for Internet of Everything (IoE) devices that are connected via multi-hop networks. Because FL algorithms hardly converge the parameters of machine learning (ML) models, this paper focuses on the convergence of ML models in function spaces. Considering that the representative loss functions of ML tasks e.g, mean squared error (MSE) and Kullback-Leibler (KL) divergence, are convex functionals, algorithms that directly update functions in function spaces could converge to the optimal solution. The key concept of this paper is to tailor a consensus-based optimization algorithm to work in the function space and achieve the global optimum in a distributed manner. This paper first analyzes the convergence of the proposed algorithm in a function space, which is referred to as a meta-algorithm, and shows that the spectral graph theory can be applied to the function space in a manner similar to that of numerical vectors. Then, consensus-based multi-hop federated distillation (CMFD) is developed for a neural network (NN) to implement the meta-algorithm. CMFD leverages knowledge distillation to realize function aggregation among adjacent devices without parameter averaging. An advantage of CMFD is that it works even with different NN models among the distributed learners. Although CMFD does not perfectly reflect the behavior of the meta-algorithm, the discussion of the meta-algorithm's convergence property promotes an intuitive understanding of CMFD, and simulation evaluations show that NN models converge using CMFD for several tasks. The simulation results also show that CMFD achieves higher accuracy than parameter aggregation for weakly connected networks, and CMFD is more stable than parameter aggregation methods.

A constrained consensus based optimization algorithm and its application to finance

In this paper, we propose a predictor-corrector type Consensus Based Optimization(CBO) algorithm on a convex feasible set. Our proposed algorithm generalizes the CBO algorithm in [11] to tackle a constrained optimization problem for the global minima of the non-convex function defined on a convex domain. As a practical application of the proposed algorithm, we study the portfolio optimization problem in finance. In this application, we introduce an objective function to choose the optimal weight on each asset in an asset-bundle, which yields the maximal expected returns given a certain level of risks. Simulation results show that our proposed predictor-corrector type model is successful in finding the optimal value.

Convergence and error estimates for time-discrete consensus-based optimization algorithms

We present convergence and error estimates of modified versions of the time-discrete consensus-based optimization (CBO) algorithm proposed in Carrillo et al. (ESAIM: Control Optim Calc Var, 2020) for general non-convex functions. In authors’ recent work (Ha et al. in Math Models Meth Appl Sci 30:2417–2444, 2020), rigorous error analysis of a modified version of the first-order consensus-based optimization algorithm proposed in Carrillo et al. (2020) was studied at the particle level without resorting to the kinetic equation via a mean-field limit. However, the error analysis for the corresponding time- discrete algorithm was not done mainly due to lack of discrete analogue of Ito’s stochastic calculus. In this paper, we provide a simple and elementary convergence and error analysis for a general time-discrete consensus-based optimization algorithm, which includes modifications of the three discrete algorithms in Carrillo et al. (2020), two of which are present in Ha et al. (2020). Our analysis provides numerical stability and convergence conditions for the three algorithms, as well as error estimates to the global minimum.

Real-time monitoring and operation of microgrid using distributed cloud–fog architecture

In this paper, a new distributed multi-agent framework based on the three layers’ fog computing architecture is developed for real-time microgrid economic dispatch and monitoring. To this end, the changes of load at any time will be tracked by the proposed technique, considering unit sudden exits and entries. Moreover, to make the system more realistic, different renewable energies, including photovoltaics (PVs), wind turbines (WTs), fuel cells (FCs), and microturbines (MT) are considered in the proposed technique. To overcome the complexity of the problem, by using advantages of fog computing, a new fast consensus-based optimization algorithm is used, which is modified based on the fuzzy adaptive leader technique. Finally, the proposed technique is simulated and tested on microgrids with 6 and 14 buses, respectively. Simulation results demonstrate and validate the effectiveness of the proposed technique, as well as the capability to track the changes of load with the interactions in real-time and the fast convergence rate.

Sampled-data self-triggered consensus-based economic dispatch problem under switching graph

To reduce the frequency of data exchange in power grid, a novel self-triggered consensus-based optimization algorithm for economic dispatch problem (EDP) is proposed for switching topologies, which guarantees that the increment cost converges to an optimal consensus value after iterations. In this brief, sampled-data framework is employed to complete the implementation of algorithm digitally. Based on that, self-triggered mechanism based approach can also avoid infinite number of communication in finite time, which can save communication cost between generators. Further, a time-varying gain function is designed to derive a tradeoff between convergence rate and triggering performance. Finally, some simulation examples are given to illustrate the effectiveness of the relevant theoretical results.

A comparative study of multi-agent control approaches for optimization of central cooling systems without significant storage

This paper presents the application of a multi-agent control methodology to large chiller plants. The approach, originally conceived to automate controller design and reduce engineering costs in building energy systems, consisted of a multi-agent simulation framework with distributed consensus-based optimization algorithms. To adapt the approach to this scenario, agents representing physical components of a cooling plant were developed and incorporated in the framework along with two alternative optimization methods: centralized, parallel optimization with a genetic algorithm (GA), and a combination of the GA with a quasi-newton method to handle non-linear equality constraints associated to physical component behavior. An existing cooling plant was utilized as case study to simulate the performance of the methods under different operating conditions. The results demonstrated the difficulty of the consensus-based algorithms to find optimal solutions. The GAs, on the other hand, showed that significant energy savings can be achieved through the implementation of multi-agent control with algorithms capable of handling non-convex objective functions and a combination of discrete and continuous variables, which are characteristic of central cooling systems.

Distributed Incremental Cost Consensus-Based Optimization Algorithms for Economic Dispatch in a Microgrid

The economic dispatch problems (EDPs) in a microgrid (MG) have been extensively investigated by a variety of emerging algorithms. In this paper, we propose two newly distributed dynamic optimization algorithms to respectively study the EDPs under both cases without and with generation constraints under a directed topology network. Two novel dynamic optimization algorithms are based on the distributed incremental cost consensus (ICC), where the mismatch between total demand and power generation is considered. Our algorithms only require the weight matrix of the directed network to be row stochastic. The theoretical analysis on the convergence of the proposed algorithms is presented by using the small gain theorem. It can be found that the algorithms are convergent at the geometric rate. Meanwhile, the power output of the generators are proved to achieve the optimal solution of EDPs based on the proposed algorithms. Finally, the corresponding conditions are also derived, and simulation studies illustrate the correctness of our results.

Distributed optimal consensus algorithms in multi-agent systems

This paper investigates the distributed optimal consensus problem by designing the multi-agent communication systems, where each agent has local access to its own strongly convex utility function. Here, we design two distributed consensus-based optimization algorithms to tackle the optimization problem of multi-agent systems. One is the second-order optimization consensus algorithm, and another is the third-order algorithm. Moreover, the two algorithms can ensure that all agents achieve consensus at a common state which minimizes the sum of the objective function under the assumption that interaction graph is strongly connected and detailed balanced. For each of the proposed algorithms, it can be found that the equilibrium point of the algorithm is the optimal point and it is proved to be asymptotically stable under given conditions. Finally, the performance of the proposed algorithms are verified by two simulations based on an IEEE 14-bus test system and an IEEE 57-bus test system.

Multiagent-Based Optimal Microgrid Control Using Fully Distributed Diffusion Strategy

This paper proposes a multiagent-based optimal microgrid control scheme using a fully distributed diffusion strategy. A two-level cooperative optimization multiagent system is adapted for distributed energy resources economic dispatch. The lower level implements an adaptive droop scheme based on online no-load frequency adjustments. The upper level implements distributed communication using diffusion between neighboring agents for optimal microgrid management. The proposed control scheme enables peer-to-peer communication among the agents without the necessity of a centralized controller, and simultaneously performs resource optimization while regulating the system frequency. The results are compared with centralized and consensus-based optimization algorithms. We have concluded that the proposed algorithm is superior over consensus algorithms in terms of convergence speed and utilizes reduced communication infrastructure compared to centralized controllers. Simulation demonstrations were conducted along with experimental results from a hardware-based microgrid using an industrial multiagent framework. The simulation and experimental results show that the proposed method and the agent framework can be deployed in real-world microgrids and offer superior decision making on optimal microgrid control.

A consensus-based model for global optimization and its mean-field limit

We introduce a novel first-order stochastic swarm intelligence (SI) model in the spirit of consensus formation models, namely a consensus-based optimization (CBO) algorithm, which may be used for the global optimization of a function in multiple dimensions. The CBO algorithm allows for passage to the mean-field limit, which results in a nonstandard, nonlocal, degenerate parabolic partial differential equation (PDE). Exploiting tools from PDE analysis we provide convergence results that help to understand the asymptotic behavior of the SI model. We further present numerical investigations underlining the feasibility of our approach.

Distributed Optimal Resource Management Based on the Consensus Algorithm in a Microgrid

A microgrid is a promising approach to provide clean, renewable, and reliable electricity by integrating various distributed generations and energy storage systems into power systems. However, highly intermittent renewable generations and various load demands pose new challenges to the optimal resource management in a microgrid. This paper proposes a fully distributed control strategy based on the consensus algorithm for the optimal resource management in an islanded microgrid. The proposed strategy is implemented through a multiagent system framework, which only requires information exchange among neighboring agents through a local network. The objective is achieved through a two-level control strategy. The upper control level is a consensus-based optimization algorithm that discovers the reference of optimal power generation or demand while maintaining the supply–demand balance. The lower control level is responsible for reference tracking of the associated component. Simulation results in the IEEE 14- and 162-bus systems are presented to demonstrate the effectiveness of the proposed control strategy.