Distributed Optimization Problem Research Articles

Distributed Adaptive Optimization Algorithm for High-Order Nonlinear Multi-Agent Stochastic Systems with Lévy Noise.

An adaptive neural network output-feedback control strategy is proposed in this paper for the distributed optimization problem (DOP) of high-order nonlinear stochastic multi-agent systems (MASs) driven by Lévy noise. On the basis of the penalty-function method, the consensus constraint is removed and the global objective function (GOF) is reconstructed. The stability of the system is analyzed by combining the generalized Itô's formula with the Lyapunov function method. Moreover, the command filtering mechanism is introduced to solve the "complexity explosion" problem in the process of designing virtual controller, and the filter errors are compensated by introducing compensating signals. The proposed algorithm has been proved that the outputs of all agents converge to the optimal solution of the DOP with bounded errors. The simulation results demonstrate the effectiveness of the proposed approach.

OPTIMAL DISTRIBUTION OF ADDITIONAL ELECTRICAL POWER

The problem of optimal distribution of additional electric power for private houses and households with solar panels or stations in adverse weather conditions over a long period of time is considered. This problem is very relevant for regions where the supply of electric power to houses and households is carried out mainly by green technology. Based on this, it is shown that when weather conditions worsen, which can last for a long time, a problem arises of providing these objects with energy allocated from backup power lines, the capacity of which is insufficient for full energy supply. To solve this problem, especially in the case of a shortage of allocated power, when it is not enough to satisfy all consumers to the maximum, the criteria for distributing the allocated limited power among houses are analyzed, examples are given for calculating the residual energy of a deep-discharge battery for each household in the event of prolonged bad weather conditions. Two options for solving the problem of optimal distribution of the allocated power are considered, taking into account the full simulation model of houses and households with electricity supply from solar panels: - distribution by residual energy taking into account the energy required to charge the batteries of private houses; - distribution by residual energy without charging the batteries. The solution to the problem of distributing limited energy between houses with solar panels is shown using the example of 5 households. For this purpose, the initial conditions and restrictions for each house are modeled. The problem of optimal distribution of allocated energy between these objects is formulated. The model of the problem itself is built and solved using the SOLVER subroutine of the Excel program. The main tables are given, in which it is necessary to enter restrictions, optimization criteria and the objective function, screenshots of the solution of this problem in two versions with the introduction of all restrictions in separate Excel windows. It is shown that solution of the issue of optimal distribution of additional energy allocated to eliminate the shortage of energy in conditions of limitation allows to the following opportunities: - provision of additional energy depending on the time of complete depletion of energy in the house prevents overloading of the electric network, incorrect openings in power lines and other unpleasant situations; - the inclusion of the module of optimal distribution of additional energy into the model during simulation, allows to save time for the seeking of additional energy and thus to increase the overall efficiency of the system. Keywords: solar panels, bad weather, additional energy, optimal distribution.

A decentralized control approach in hypergraph distributed optimization decomposition cases

We study the main decomposition approaches (primal, dual and primal–dual) for a distributed optimization problem from a dynamical system perspective where the couplings among the variables of the optimization problem are described by an undirected, unweighted hypergraph. We conduct stability analysis for the respective dynamical systems of the decomposition cases by using non linear decentralized control theoretic techniques and spectral properties of the respective communication matrices, i.e., the incidence and the Laplacian matrices of the hypergraph. Finally, we provide numerical simulations under a specific coalitional setting that demonstrate the superiority of the hypergraph compared to its respective graph analogue, the clique expansion graph, for the given decomposition algorithms in terms of convergence rate and information transmission efficiency.

Distributed zero-gradient-sum optimisation algorithm with an edge-based adaptive event-triggered mechanism

This paper addresses the continuous-time distributed optimisation problem over networks, where the global objective function is formed by a sum of convex local objective functions. To avoid continuous communication among agents, a distributed adaptive Zero-Gradient-Sum (ZGS) optimisation algorithm under a dynamic event-triggered scheme is proposed. This is achieved by dynamically adjusting the coupling strengths of adjacent agents within the network. Our analysis confirms that the proposed algorithm will exponentially converge to the optimal solution provided that the underlying communication graph is undirected and connected. Additionally, we demonstrate that our event-triggered scheme is not subject to Zeno behaviour, which is a theoretical concern in systems with frequent event triggers.

A Collaborative Neurodynamic Optimization Approach to Distributed Chiller Loading.

In this article, we present a collaborative neurodynamic optimization approach to distributed chiller loading in the presence of nonconvex power consumption functions and binary variables associated with cardinality constraints. We formulate a cardinality-constrained distributed optimization problem with nonconvex objective functions and discrete feasible regions, based on an augmented Lagrangian function. To overcome the difficulty caused by the nonconvexity in the formulated distributed optimization problem, we develop a collaborative neurodynamic optimization method based on multiple coupled recurrent neural networks reinitialized repeatedly using a meta-heuristic rule. We elaborate on experimental results based on two multi-chiller systems with the parameters from the chiller manufacturers to demonstrate the efficacy of the proposed approach in comparison to several baselines.

Distributed stochastic optimization algorithm with non-consistent constraints in time-varying unbalanced networks

This study focuses on a class of distributed optimization problems with non-consistent local set constraints in time-varying unbalanced networks. Considering that each agent only has access to local gradient information with random errors, this paper proposes a distributed stochastic gradient projection algorithm. Under some conditions of random errors and a uniformly joint strongly connected topology, it is shown that the local decision states of all agents converge to a common optimal solution with a probability of one, achieving a sublinear convergence rate of O(lnk/k). Notably, under the condition that at least one local function exhibits strong convexity, the algorithm achieves a faster sublinear convergence rate of O(1/k). These theoretical results are validated through numerical simulations. Furthermore, this paper applies the proposed algorithm to solve for the unknown parameters in distributed linear regression problems with incomplete data. Numerical results indicate that the algorithm not only aligns with the convergence properties reported in existing literature but also demonstrates significant superiority in convergence speed. This important discovery not only lays a solid theoretical foundation for the further application and extension of the algorithm but also reveals its immense potential in solving practical problems.

A resilient control algorithm to distributed optimization under DoS attacks

SummaryThis article focuses on a distributed optimization problem under an undirectd and connected network affected by DoS attacks. First, new event‐triggered mechanisms are developed to schedule the communication between agents. Then, an event‐based resilient distributed optimization algorithm that removes the requirement of continuous communication in the existing results is designed to achieve consensus optimization. Assisted by the design of regulator equations and the algebraic graph theory, it proves that the proposed algorithm is exponentially convergent. Moreover, Zeno behavior can be excluded by reductio ad absurdum. The simulation results validate that the proposed algorithm is effective.

Distributed model predictive control of fan coil system

To address the issues of large overshoot, long system stabilization time, and slow response associated with traditional PID-based and centralized model predictive control (MPC) of fan coil systems, this study proposes a distributed model predictive control (DMPC) method suitable for multi-zone building temperature regulation. Specifically, based on the thermal dynamic model of different zones, and aiming to constrain indoor temperature within set limits and minimize water system pressure drop and fan coil speed, a distributed MPC architecture for fan coil systems is developed. To solve the distributed optimization problem, the Nash equilibrium is introduced to find the global balance point within the distributed structure, and the Beetle Antennae Search-Particle Swarm Optimization (BAS-PSO) hybrid algorithm is used for online optimization of local MPCs. Experimental results show that compared to traditional PID control, the distributed MPC achieves coordinated multi-zone indoor temperature regulation with better dynamic performance, with the average absolute temperature error in each zone below 5 % and an energy savings rate of 16.23 %. Through semi-physical simulation, it is demonstrated that the proposed distributed MPC method has better applicability when the number of fan coils exceeds 10, compared to centralized MPC.

Stable Matching Algorithm Approach to Resolving Institutional Projects Allocation and Distribution Optimization Problems

Stable Matching Algorithm Approach to Resolving Institutional Projects Allocation and Distribution Optimization Problems

Privacy preserving distributed event-triggered optimisation for multi-agent systems

The distributed optimisation problem with privacy-preserving properties is considered in this paper. To solve this problem, a zero-gradient-sum algorithm based on output mask is proposed. An event-triggered condition is designed by using the output mask, which reduces the communication burden of the system effectively. The theoretical results show that the proposed algorithm can solve the optimisation problem, while the privacy information of nodes is preserved. In addition, the event-triggered condition designed based on the output mask can effectively avoid Zeno behaviour. Two simulation cases are performed to validate the effectiveness of the algorithm.

Delay effects on distributed constrained optimization over double‐integrator multi‐agent systems

AbstractCommunication plays a pivotal role in distributed optimization problems, where unavoidable communication delays are presented. This research studies the distributed constrained optimization problem concerning second‐order multi‐agent systems with double‐integrator under time‐varying communication delays. An adaptive distributed optimization algorithm using multi‐agent system consensus technique and Karush–Kuhn–Tucker conditions is developed to deal with this problem. The local constraint term is solved adaptively through local dual Lagrange multipliers. When the cost function is strongly convex, and the communication topology is undirected and connected, we employ the Lasalle invariance principle to analyze the delay effects on convergence analysis. Moreover, we give an upper bound on communication delay. Finally, the provided numerical simulation examples demonstrate that the developed method is robust for the limited communication delay and the derived results are conservative.

Robust topology and discrete fiber orientation optimization under principal material uncertainty

This paper introduces a formulation of the robust topology optimization problem that is tailored for designing fiber-reinforced composite structures with spatially varying principal mechanical properties. Specifically, a methodology is developed that incorporates the spatial variability in the engineering constants of the composite lamina into the concurrent topology (i.e., material distribution) and morphology (i.e., fiber orientation distribution) optimization problem for the minimization of the robust compliance function. The spatial variability in the mechanical properties of the lamina is modeled as a homogeneous random field within the design domain by means of the Karhunen-Loe´ve series expansion, and is thereafter intrusively propagated into the stochastic finite element analysis of the composite structure. To carry out the stochastic finite element analysis per iteration of the optimization cycle, the first-order perturbation method is utilized for approximating the current state variables of the physical system. The resulting robust topology and fiber orientation optimization problem is formulated step-by-step for the minimization of the robust compliance function. With the view of solving the optimization problem at hand by means of gradient-based solution algorithms, the first-order derivatives of the involved design functions w.r.t. the associated design variables are analytically derived. The present work concludes with a series of numerical examples, focusing on the benchmark academic case studies of the 2D cantilever and the half part of the Messerschmitt-Bölkow-Blohm beam, aiming to demonstrate the developed methodology as well as to explore the effect that different parameterization instances of the random field bear on the predicted topology and morphology of the beams.

A multi-period optimal distribution model of emergency resources for responding to COVID-19 under uncertain conditions

Ideally, optimal emergency resource allocation would have been vital for effective relief work during the COVID-19 outbreak. However, the suddenness of the epidemic and uncertainty of its spread added some difficulties to distributing emergency resources. First, this study introduces triangular fuzzy numbers to describe the uncertainty of supply and demand of emergency resources, and interval numbers to describe the time required for resource transportation under disaster conditions. To minimize the total delivery time and difference in the total satisfaction rate, this study constructs an optimal model for emergency resource distribution under uncertain conditions that considers both efficiency and equity. Subsequently, an improved genetic algorithm (IMGA) is proposed to obtain the optimal decision scheme. Finally, a case study on emergency resource distribution during the COVID-19 pandemic is conducted for model verification. The results demonstrate that the proposed model can improve the efficiency and effect of emergency resource distribution. The model allocates some emergency resources to each demand site during each emergency period, which can help avoid large losses caused by extreme shortages of resources at a certain demand point. The emergency resource allocation scheme considers the transportation time and degree of impact, which is beneficial for enhancing the flexibility of decision-making and practical applicability of distribution operations. A comparative analysis of the algorithms shows that the proposed IMGA is an effective method for managing emergency resource distribution optimization problems because it has higher solving efficiency, better convergence, and stronger stability. These findings can provide decision support for the optimal distribution of large-scale, multiperiod emergency resources during the COVID-19 pandemic.

Distributed finite-time optimization for networked Euler–Lagrange systems under a directed graph

In this paper, we investigate the finite-time distributed optimization problem for multiple Euler–Lagrange systems. A new distributed optimization control scheme is presented to achieve the state agreement in finite time while minimizing the sum of each agent’s local cost function. The proposed algorithm has the advantage of being able to achieve distributed finite-time optimization consensus under general unbalanced connected directed communication graphs. By virtue of finite-time Lyapunov theory and convex optimization, the finite-time convergence for the algorithm is analyzed. A numerical example is also presented to illustrate the effectiveness of the obtained results.

Кількісне оцінювання функціональної ефективності мережевих структур за аналізом чутливості оптимізаційної моделі до зовнішніх впливів

The real network organizational structure is constantly under the influence of external influences, which can cause unexpected consequences due to the cascading spread of relevant disturbances directed at critically important parameters. Parametric sensitivity analysis is the latest computational procedure, the results of which provide the researcher with an idea and quantitative assessment of the appropriate response of the structure in the form of an analytical platform as a composition and interaction of scenario analysis, cascade process development and optimization modeling technology. The main task is timely adaptation to expected changes in business analytics, oriented to optimal resources distribution. The generalized minimum cost problem is a typical example of widespread problems of optimal distribution and use of limited, in particular, energy resources, of any nature and purpose, using the scientific management and analysis of decisions, in its network model attention is paid to the potentials of nodes. The maximum flow problem is an example of the active and effective application of methods and models of flows optimization, where the processes and objects used a network organization, the energy systems and complexes investigated, where arc parameters are critical in its model. In the article, for these classic optimization problems, mathematical and spreadsheet models were built on concrete examples. Direct and dual mathematical programming problems were solved, the following stages were carried out - the organization of a computer experiment and the construction of an analytical platform for evaluating the behavior of the network structure under the influence of external influences and disturbances as their consequences. To take into account the specific conditions of real objects with a network structure at the stage of modification, the model supplemented with appropriate restrictions and correction of input data sets. The obtained results should be useful for planning and management personnel and top-management for discussion and decision-making regarding, for example, the optimal placement of energy equipment, the weighted distribution of energy flows in the "source-sink" system or the appointment of executors for work, automated project management using project networks, etc. Keywords: optimization modeling, maximum network flow, minimum cost flows, network organizational structures, decision making, spreadsheet modeling and analytics.

Fast distributed co-optimization of electricity and natural gas systems hedging against wind fluctuation and uncertainty

The synergistic operation of integrated electricity and natural gas systems (IEGS) in the presence of wind power necessitates a distributed optimization framework that ensures information privacy. However, incorporating wind power penetration leads to distributed optimization problems including uncertainty and unstable distributed algorithm’s convergence. With the increasing proportion of wind power in IEGS, fluctuating wind power penetration highly affects the convergence of distributed optimization solutions, resulting in uncontrollable optimization time of the distributed algorithm. Accordingly, this paper investigates the impact of wind fluctuation and uncertainty on distributed IEGS optimization and proposes a novel fast distributed co-optimization framework. Specifically, an adaptive alternating direction method of multipliers (ADMM) is developed to accommodate wind fluctuation. Based on designed rules, the penalty parameter is updated in every step to maximize the gradient of the optimization objective. The experiments are conducted using real wind power generation data sourced from a wind farm in Australia and a classical IEGS framework composed of the IEEE 24-bus electricity system and a 12-node natural gas system. Compared to original ADMM and residual balancing ADMM, the proposed framework achieves an average reduction of 0–91.4% in the number of iterative steps for multiple solution iterations across various scenarios, with a corresponding decrease in the standard deviation by 13.8%–93.0%.

Distributed optimization for discrete time‐varying linear multi‐agent systems with event‐triggered communication

AbstractThis paper studies the distributed optimization problem (DOP) of discrete time‐varying linear multi‐agent systems (MASs), in which the global objective function is formed by a sum of local convex objective functions. Firstly, a DOP with discrete time‐varying MASs is considered, in which the time‐varying linear matrix satisfies a certain equality constraint. To solve this problem, a novel discrete‐time distributed optimization algorithm (DOA) with event‐triggered communication mechanism (ETCM) is proposed. Secondly, by constructing the error dynamical system and using a series of inequality techniques, some sufficient conditions for achieving consensus and obtaining the optimal solution are established. It is found that the considered MAS has generality and the proposed DOA has the advantage of reducing communication burden. Finally, a numerical simulation is presented to verify the validity of theoretical results.

Output feedback distributed optimization algorithms of higher‐order uncertain nonlinear multi‐agent systems

AbstractOutput feedback distributed optimization problem is studied for higher‐order multi‐agent systems with uncertain nonlinearities, which are assumed to satisfy linear growth conditions. The agents dynamics are permitted to be heterogenous with different nonlinearities. The problem is solved by embedded control approach based output feedback distributed optimization algorithms. First, a first‐order optimal signal generator is designed, of which outputs reach the minimizer of the global cost function. Second, embedding the generator into the feedback loop and taking the outputs of the generator as the reference outputs of the agents, output feedback tracking controllers are designed for the higher‐order multi‐agent systems on the basis of nonseparation principle and feedback domination technique. Under the proposed control algorithms, all the agents outputs asymptotically approach a bounded neighbor region of the global minimizer. Simulations demonstrate the effectiveness of the proposed output feedback distributed optimization algorithms.

Prescribed-time distributed optimization problem with constraints

In recent years, distributed optimization problem have a wide range of applications in various fields. This paper considers the prescribed-time distributed optimization problem with/without constraints. Firstly, we assume the state of each agent is constrained, and the prescribed-time distributed optimization algorithm with constraints is designed on the basis of gradient projection algorithm and consensus algorithm. Secondly, the constrained distributed optimization problem is transformed into the unconstrained distributed optimization problem, and according to the gradient descent algorithm and consensus algorithm, we also propose the prescribed-time distributed optimization algorithm without constraints. By designing the appropriate objective functions, we prove the multi-agent system can converge to the optimal solution within any prescribed-time, and the convergence time is fully independent of the initial conditions and system parameters. Finally, three simulation examples are provided to verify the validity of the designed algorithms.

Distributed short-term predictive control for AUV clusters in underwater cooperative hunting tasks

A distributed short-term predictive control method is proposed to be applied to autonomous underwater vehicle (AUV) clusters for underwater cooperative hunting tasks. To support AUV clusters in unknown underwater dynamic environments, the constraints of information acquisition, AUV motion constraints, collision avoidance, and smoothness requirements of motion paths in the tasks are considered. First, a distributed multi-constraint optimization model including target attraction, interactions between AUVs, continuous obstacle avoidance, and control effort is proposed. Effective information between AUVs is extended under limited information conditions. Second, each AUV solves the distributed optimization problem and determines the desired trajectory in real time using information from the prediction horizon. In the simulation experiments, the proposed method is demonstrated to provide at least 12.19%, 11.08%, and 79.89% improvement in terms of task completion time, energy consumption, and path smoothing, respectively, when compared to other methods under the same information conditions. The robustness of the proposed method under different noise conditions is verified by positioning noise experiments, and the task can still be accomplished even if the noise level reaches 75% of the safe distance. The adaptability of the proposed method is demonstrated by simulations in different environments.