Due to the increase in the advances in wireless communication, there has been an increase in the use of multi-agents systems to complete any given task. In various applications, multi-agent systems are required to solve an underlying optimization problem to obtain the best possible solution within a feasible region. Solving such multi-agent optimization problems in a distributed framework preferable over centralized frameworks as the former ensures scalability, robustness, and security. Further distributed optimization problem becomes challenging when the decision variables of the individual agents are coupled. In this thesis, a distributed optimization problem with coupled constraints is considered, where a network of agents aims to cooperatively minimize the sum of their local objective functions, subject to individual constraints. This problem setup is relevant to many practical applications like formation flying, sensor fusion, smart grids, etc. For practical scenarios, where agents can solve their local optimal solution efficiently and require fewer assumptions on objective functions, the Alternating Direction Method of Multipliers(ADMM)-based approaches are preferred over gradient-based approaches. For such a constraint coupled problem, several distributed ADMM algorithms are present that guarantee convergence to optimality but they do not discuss the complete analysis for the rate of convergence. Thus, the primary goal of this work is to improve upon the convergence rate of the existing state-of-the-art Tracking-ADMM (TADMM) algorithm to solve the above-distributed optimization problem. Moreover, the current analysis in literature does not discuss the convergence in the case of a time-varying communication network. The first part of the thesis focuses on improving the convergence rate of the Tracking-ADMM algorithm to solve the above-distributed optimization problem more efficiently. To this end, an upper bound on the convergence rate of the TADMM algorithm is derived in terms of the weight matrix of the network. To achieve faster convergence, the optimal weight matrix is computed using a semi-definite programming (SDP) formulation. The improved convergence rate of this Fast-TADMM (F-TADMM) is demonstrated with a simple yet illustrative, coupled constraint optimization problem. Then, the applicability of F-TADMM is demonstrated to the problem of distributed optimal control for trajectory generation of aircraft in formation flight. In the second part of the thesis, the convergence analysis for TADMM is extended while considering a time-varying communication network. The modified algorithm is named as Time-Varying Tracking (TV-TADMM). The formal guarantees on asymptotic convergence are provided with the help of control system analysis of a dynamical system that uses Lyapunov-like theory. The convergence of this TV-TADMM is demonstrated on a simple yet illustrative, coupled constraint optimization problem with switching topology and is compared with the fixed topology setting.

Cette thèse s’intéresse au problème d’estimation et d’optimisation distribuée dans les réseaux asynchrones, c’est à dire en n’utilisant que des communication locales et asynchrones. A partir de multiples applications allant de l’apprentissage automatique aux réseaux de capteurs sans-fils, nous concevons et analysons théoriquement de nouveaux algorithmes résolvant trois problèmes de nature très différentes : la propagation de la plus grande des valeurs initiales, l’estimation de leur moyenne et enfin l’optimisation distribuée. / This thesis addresses the distributed estimation and optimization of a global value of interest over a network using only local and asynchronous (sometimes wireless) communications. Motivated by many different applications ranging from cloud computing to wireless sensor networks via machine learning, we design new algorithms and theoretically study three problems of very different nature : the propagation of the maximal initial value, the estimation of their average and finally distributed optimization.

En radioastronomie, les signaux d'intérêt mesurés par les interféromètres sont perturbés par de nombreux effets environnementaux et instrumentaux, nécessitant la mise en œuvre de techniques algorithmiques pour les traiter et pouvoir ainsi reconstruire in fine des images parfaitement nettes de l'espace. Cette étape de correction des perturbations se nomme la calibration et repose généralement sur une modélisation gaussienne du bruit, pour une seule fréquence considérée. Cependant, en pratique, cette l'hypothèse n'est pas toujours valide car de multiples sources inconnues à faible intensité sont visibles dans le champ de vision et des interférences radioélectriques perturbent les données. En outre, réaliser une calibration indépendante, fréquence par fréquence, n'est pas la manière la plus optimale de procéder. Le but de ce travail est donc de développer des algorithmes de correction dans le traitement des signaux radio qui soient robustes à la présence d'éventuelles valeurs aberrantes ou sources d'interférences, et qui soient adaptés au contexte multi-fréquentiel. Par conséquent, nous nous appuyons sur une modélisation plus générale que la loi gaussienne, appelé processus Gaussien composé, et proposons un algorithme itératif basé sur l'estimation au sens du maximum de vraisemblance. En accord avec le scénario multi-fréquentiel sous étude, nous exploitons la variation spectrale des perturbations en utilisant des méthodologies telles que l'optimisation distribuée sous contraintes et le traitement parallèle des données. / Accurate calibration is of critical importance for new advanced interferometric systems in radio astronomy in order to recover high resolution images with no distortions. This process consists in correcting for all environmental and instrumental effects which corrupt the observations. Most state-of-the-art calibration approaches assume a Gaussian noise model and operate mostly in an iterative manner for a mono-frequency scenario. However, in practice, the Gaussian classical noise assumption is not valid as radio frequency interference affects the measurements and multiple unknown weak sources appear within the wide field-of-view. Furthermore, considering one frequency bin at a time with a single centralized agent processing all data leads to suboptimality and computational limitations. The goal of this thesis is to explore robustness of calibration algorithms w.r.t. the presence of outliers in a multi-frequency scenario. To this end, we propose the use of an appropriate noise model, namely, the so-called coumpound-Gaussian which encompasses a broad range of different heavy-tailed distributions. To combine limited computational complexity and quality of calibration, we designed an iterative calibration algorithm based on the maximum likelihood estimator under the compound-Gaussian modeling. In addition, a computationally efficient way to handle multiple sub-frequency bands is to apply distributed and decentralized strategies. Thus, the global operational load is distributed over a network of computational agents and calibration amounts to solve a global constrained problem thanks to available variation models or by assuming smoothness across frequency.

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In the intersection between machine learning, artificial intelligence and mathe- matical computation lies optimization. A powerful tool that enables us to solve a variety of large scale problems. The purpose of this work is to explore optimiza- tion in the distributed setting. We will then touch on factors that contribute to a faster and more stable algorithm while solving a distributed optimization problem. The main factor we will look into is how we can integrate control.

Smart grid is more than just the smart meters. The future smart grids are expected to include a high penetration of distributed generations (DGs), most of which will consist of renewable energy sources, such as solar or wind energy. It is believed that the high penetration of DGs will result in the reduction of power losses, voltage profile improvement, meeting future load demand, and optimizing the use of non-conventional energy sources. However, more serious problems will arise if a decent control mechanism is not exploited. An improperly managed high PV penetration may cause voltage profile disturbance, conflict with conventional network protection devices, interfere with transformer tap changers, and as a result, cause network instability. Indeed, it is feasible to organize DGs in a microgrid structure which will be connected to the main grid through a point of common coupling (PCC). Microgrids are natural innovation zones for the smart grid because of their scalability and flexibility. A proper organization and control of the interaction between the microgrid and the smartgrid is a challenge. Cooperative control makes it possible to organize different agents in a networked system to act as a group and realize the designated objectives. Cooperative control has been already applied to the autonomous vehicles and this work investigates its application in controlling the DGs in a micro grid. The microgrid power objectives are set by a higher level control and the application of the cooperative control makes it possible for the DGs to utilize a low bandwidth communication network and realize the objectives. Initially, the basics of the application of the DGs cooperative control are formulated. This includes organizing all the DGs of a microgrid to satisfy an active and a reactive power objective. Then, the cooperative control is further developed by the introduction of clustering DGs into several groups to satisfy multiple power objectives. Then, the cooperative distribution optimization is introduced iii to optimally dispatch the reactive power of the DGs to realize a unified microgrid voltage profile and minimize the losses. This distributed optimization is a gradient based technique and it is shown that when the communication is down, it reduces to a form of droop. However, this gradient based droop exhibits a superior performance in the transient response, by eliminating the overshoots caused by the conventional droop. Meanwhile, the interaction between each microgrid and the main grid can be formulated as a Stackelberg game. The main grid as the leader, by offering proper energy price to the micro grid, minimizes its cost and secures the power. This not only optimizes the economical interests of both sides, the microgrids and the main grid, but also yields an improved power flow and shaves the peak power. As such, a smartgrid may treat microgrids as individually dispatchable loads or generators.

In the past 10 years, the growth of machine learning technology has been significant, largely due to the availability of large datasets for training. However, gathering a sufficient amount of data on a central server can be challenging. Additionally, with the rise of mobile networking and the large amounts of data generated by IoT devices, privacy and security issues have become a concern, resulting in government regulations such as GDPR, HIPAA, CCPA, and ADPPA. Under these circumstances, traditional centralized machine learning methods face a problem in that sensitive data must be kept locally for privacy reasons, making it difficult to achieve the desired learning outcomes. Federated learning (FL) offers a solution to this by allowing for a global shared model to be trained by exchanging locally computed optimums instead of sharing the actual data. Despite its success as a natural solution for IoT machine learning implementation, Federated learning (FL) still faces challenges with regards to security and performance. These include high communication costs between IoT devices and the central server, the potential for sensitive information leakage and reduced model precision due to the aggregation process in the distributed IoT network, and performance concerns caused by the heterogeneity of data and devices in the network. In this dissertation, I present practical and effective techniques with strong theoretical supports to address these challenges. To optimize communication resources, I introduce a new multi-server FL framework called MS-FedAvg. To enhance security, I propose a robust defense algorithm called LoMar. To address data heterogeneity, I present FedLGA, and for device heterogeneity, I propose FedSAM.

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In the first part of this thesis, we propose a distributed consensus algorithm under multi-layer multi-group structure with communication time delays. It is proven that the consensus will be achieved in both time-varying and fixed communication delays. In the second part, we study the distributed optimization problem with a finite-time mechanism. It is shown that our distributed proportional-integral algorithm can exponentially converge to the unique global minimizer when the gain parameters satisfy the sufficient conditions. Moreover, we equip the proposed algorithm with a decentralized algorithm, which enables an arbitrarily chosen agent to compute the exact global minimizer within a finite number of time steps, using its own states observed over a successive time steps. In the third part, it is shown the implementation of accelerated distributed energy management for microgrids is achieved. The results presented in the thesis are corroborated by simulations or experiments.

Electric power grids are undergoing major changes mainly driven by integrating high levels of renewable energy, advancing energy storage technologies, and deploying electric vehicles at scale. Along with numerous benefits offered by this grid modernization, technical challenges also arise largely due to variability and uncertainty of renewable energy resourcesas well as the stochastic nature of electric transportation charging demand. This thesis intends to cater the need of such modern power systems by developing a new computationally efficient function space-based distributed solution methodology for continuous-time optimal power flow (OPF) and unit commitment (UC) problems in multiarea power system. This solution methodology serves as the enhanced operation tool to enable continuous-time power exchange between interconnected systems, leveraging the operational flexibility unlocked through higher-fidelity modeling, while preserving data privacy among the participating areas. The core formulation of the proposed methodology is divided into two chapters, addressing the solutions for OPF and UC problems, respectively. Both solution methodology for OPF and UC problems amalgamate the unique properties of variational optimization, function space representation, and appropriate distributed algorithms, alternating direction method of multipliers (ADMM) and analytical target cascading (ATC), respectively to enable continuous-time power exchange between adjacent areas. At first, the centralized multi-area OPF and UC problems are formulated as variational optimization problems with continuous-time load and decision variables (power generation, voltage phase angles, line/ tieline power flows), which is then converted to a conventional optimization problem by projecting the load and decision trajectories into the Bernstein function space. While the OPF is formulated as a Quadratic Programming (QP) with linear and quadratic constraints to represent the transmission network, UC problem includes more involved formulation with consideration for additional energy storage and is modeled as a Multi-Integer Linear Programming (MILP) to incorporate binary commitment variables of the generating units. The next step involves decomposing the centralized formulation to individual sub-problems of individual areas using distributed algorithm- ADMM and ATC, which are selected based on careful analysis of references from related works.The developed solution methodology for OPF and UC problems are then implemented on a synthesized three-area network and the three-area IEEE Reliability Test System (IEEE-RTS) respectively. The numerical results are presented as different cases to highlight the performance of the proposed methods in terms of solution accuracy and achieving optimal decisions on interconnection power exchange such that the energy and ramping needs of areas are met in both OPF and UC problems.