Distributed Optimization Problem Research Articles

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.

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%.

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.

A Second-Order Primal–Dual Dynamics for Set Constrained Distributed Optimization Problems

A Second-Order Primal–Dual Dynamics for Set Constrained Distributed Optimization Problems

Two-stage problems of optimal location and distribution of the humanitarian logistics system’s structural subdivisions

Purpose. To ensure the rational organization of the evacuation of people from a region affected by an emergency by developing a mathematical and algorithmic toolkit that will allow for the early distribution of transport and material resources, maximizing coverage of the affected areas while minimizing evacuation time. Methodology. System analysis of evacuation processes; mathematical modeling, the theory of continuous problems of optimal partitioning of sets, non-differentiable optimization. Findings. The object of the study is the two-stage evacuation logistic processes that occur when serving the population of areas affected by emergencies of a natural or technogenic nature. The research considers the possibility of optimally distributing human flows within the transportation system, the structural subdivisions of which are first-stage centers (first aid stations that carry out the reception of citizens from areas affected by the disaster) and second-stage centers (specialized units of the emergency aid system that provide further services to the evacuated population). The proposed mathematical model deals with the problem of optimally partitioning a continuous set with the placement of subset centers and additional connections. Methods for its solution have been described. We demonstrate the versatility of these models, as they can be used to describe logistic evacuation processes, organize assembly points, intermediate locations, evacuation reception points, and those providing primary assistance to the affected population. We calculate the appropriate number of essential products and deliver them from existing warehouses through distribution centers to the affected areas. Originality. As preventive measures to increase the level of population safety during an emergency, we consider the optimal placement of rescue facilities and the zoning of the territory to distribute evacuation traffic. We also address the problem of the optimal distribution of human flows in the transport and logistics system. Practical value. The presented models, methods, and algorithms enable the solution of many practical problems related to the development of preventive measures and the planning of rescue operations to ensure the population’s safety in case of emergencies. The theoretical results obtained are translated into specific recommendations that can be utilized when addressing logistical problems related to the organization of primary evacuation of the population from affected areas and their subsequent transportation to safer locations for further assistance.

Resilient event‐triggered fault‐tolerant optimization for Euler–Lagrange plants with intermittent communication and asynchronous DoS attacks

AbstractThis paper investigates the distributed optimization problem of networked Euler–Lagrange (EL) systems with actuator faults and unknown model parameters. In order to solve this issue, some event‐triggered fault‐tolerant (ETFT) optimal coordination algorithms with the mechanism of intermittent communication and intermittent controller update are designed for the first time to compensate for the impact of physical faults on the system, and reduce communication costs and control resource consumption. For practical applications, it is difficult for the system to have a secure environment if there exist malicious attacks blocking network channels among agents. Therefore, the resilient ETFT optimal coordination with intermittent communication subject to asynchronous denial‐of‐service (DoS) attacks is considered. Especially, auxiliary systems with adaptive gains are introduced in the algorithms due to the unavailability of the EL model parameters, which also enable virtual data exchange between agents and protect actual state information. Note that the algorithms are fully distributed as the global topology information is not required. Finally, the feasibility of the proposed algorithms is verified through numerical simulation of examples.

Distributed Lyapunov-Based Model Predictive Control for AUV Formation Systems with Multiple Constraints

This paper focuses on the formation tracking issue of autonomous underwater vehicles (AUVs) subject to multiple constraints in three-dimensional space. We developed a novel distributed Lyapunov-based model predictive controller (DLMPC) with a fast finite-time extended state observer (FFTESO). Initially, the external disturbances and internal uncertainties of each AUV were precisely compensated using the designed FFTESO. Subsequently, we proposed DLMPC-based position tracking and velocity tracking controllers, which solved an online optimization problem to determine optimal velocities and control forces. This hierarchical framework effectively managed system constraints, such as state constraints and actuator saturation. Additionally, the Lyapunov-based backstepping control law was applied to construct stability constraints in the distributed optimization problem, ensuring the recursive feasibility and closed-loop system stability of the proposed scheme. Sufficient conditions and attraction regions to ensure stability were explicitly provided. Finally, the simulation results demonstrated that the proposed method improved both the convergence speed and tracking accuracy by at least 30% compared to other methods.

Distributed Subgradient Method With Random Quantization and Flexible Weights: Convergence Analysis.

The distributed subgradient (DSG) method is a widely used algorithm for coping with large-scale distributed optimization problems in machine-learning applications. Most existing works on DSG focus on ideal communication between cooperative agents, where the shared information between agents is exact and perfect. This assumption, however, can lead to potential privacy concerns and is not feasible when wireless transmission links are of poor quality. To meet this challenge, a common approach is to quantize the data locally before transmission, which avoids exposure of raw data and significantly reduces the size of the data. Compared with perfect data, quantization poses fundamental challenges to maintaining data accuracy, which further impacts the convergence of the algorithms. To overcome this problem, we propose a DSG method with random quantization and flexible weights and provide comprehensive results on the convergence of the algorithm for (strongly/weakly) convex objective functions. We also derive the upper bounds on the convergence rates in terms of the quantization error, the distortion, the step sizes, and the number of network agents. Our analysis extends the existing results, for which special cases of step sizes and convex objective functions are considered, to general conclusions on weakly convex cases. Numerical simulations are conducted in convex and weakly convex settings to support our theoretical results.

Distributed Computation for Sparse Recovery via Continuous-Time Neurodynamic Approach

Sparsity has been extensively employed in multimedia sensing and computing in consumer electronics, signal and image processing, depth video codec, adaptive sparse-type equalizer, blind speech separation, and machine learning. Throughout this paper, we propose a novel distributed projection neurodynamic approach for solving the Basis Pursuit (BP) with flexible partition methods in a distributed manner. The proposed neurodynamic approach requires only that the network is undirected and connected, and no node can access the entire matrix simultaneously. First, we equivalently formulate the BP into a standard distributed optimization problem with a flexible partition-by-blocks method to obtain global information, and discuss the equivalence of their optimality conditions. Then, we propose a distributed continuous-time neurodynamic approach on the basis of primal-dual dynamical systems and projection operators, and also study its global convergence property. Finally, numerical experiments on sparse signals and image recovery further verify the effectiveness and superiority of our proposed neurodynamic approach.

Privacy-Preserving Distributed ADMM With Event-Triggered Communication.

This article addresses distributed optimization problems, in which a group of agents cooperatively minimize the sum of their private objective functions via information exchanging. Building on alternating direction method of multipliers (ADMM), we propose a privacy-preserving and communication-efficient decentralized quadratically approximated ADMM algorithm, termed PC-DQM, for solving such type of problems under the scenario of limited communication. In PC-DQM, an event-triggered mechanism is designed to schedule the communication instants for reducing communication cost. Simultaneously, for privacy preservation, a Hessian matrix with perturbed noise is introduced to quadratically approximate the objective function, which results in a closed form of primal vector update and then avoids solving a subproblem at each iteration with possible high computation cost. In addition, the triggered scheme is also utilized to schedule the update of Hessian, which can also reduce computation cost. We theoretically show that PC-DQM can protect privacy but without losing accuracy. In addition, we rigorously prove that PC-DQM converges linearly to the exact optimal solution for strongly convex and smooth objective functions. Finally, numerical simulation is presented to illustrate the effectiveness and efficiency of our algorithm.

A Specified-Time Convergent Multiagent System for Distributed Optimization With a Time-Varying Objective Function

This technical note presents a specified-time convergent multi-agent system for distributed optimization with a time-varying objective function subject to equality constraints. Different from the static optimal solutions to most existing distributed optimization problems, the optimal solutions are time-varying due to the time-varying objective function in this problem. A distributed protocol law is designed to ensure all the agents' convergence to feasible and suboptimal solutions within a specified settling time and keep tracking the time-dependent optimal solutions. The specified-time convergence of the system and the asymptotic optimality of the solution generated by the system are proved based on the Lyapunov theory. A salient feature of the multi-agent system is that its upper bound of settling time can be specified in advance. Two examples are presented to illustrate the theoretical results.

Nabla fractional distributed optimization algorithms over undirected/directed graphs

The primary focus of this paper centers on investigating unconstrained distributed optimization problems over undirected or directed graphs. One noteworthy departure from current distributed optimization algorithms in the continuous-time domain is the integration of nabla fractional calculus, which augments algorithmic performance by reducing iterative complexity. Through rigorous analysis, this paper demonstrates that the two algorithms presented converge at the Mittag–Leffler rate to the precise solution of a distributed optimization problem over a connected undirected graph or a connected balanced directed graph with strongly convex and smooth objective functions. The research findings provide valuable insights into the potential utility of nabla fractional calculus in distributed optimization problems, highlighting the possibility of enhancing the efficiency and effectiveness of distributed optimization algorithms.

A Generic Algorithm Framework for Distributed Optimization Over the Time-Varying Network With Communication Delays

In this paper, we study the distributed optimization problem (DOP) over the time-varying communication network topology with communication delays. The existing distributed optimization algorithms (DOAs) cannot be directly applied to DOPs over the time-varying network with communication delays. In this paper, by adding new virtual agents and redefining communication links, some delay properties are realized and existing DOAs can then be reused to solve the DOPs. Thus, a generic algorithm framework is established which can apply to many important DOAs, such as the mirror descent algorithm and dual averaging algorithm, to solve DOPs over the time-varying network with communication delays. The convergence results of the DOAs are established and verified under the generic algorithm framework.