Loris Cannelli (6933851)
13 August 2019
<div>The focus of this Dissertation is to provide a unified and efficient solution method for an important class of nonconvex, nonsmooth, constrained optimization problems. Specifically, we are interested in problems where the objective function can be written as the sum of a smooth, nonconvex term, plus a convex, but possibly nonsmooth, regularizer. It is also considered the presence of nonconvex constraints. This kind of structure arises in many large-scale applications, as diverse as information processing, genomics, machine learning, or imaging reconstruction.</div><div></div><div>We design the first parallel, asynchronous, algorithmic framework with convergence guarantees to stationary points of the class of problems under exam. The method we propose is based on Successive Convex Approximation techniques; it can be implemented with both fixed and diminishing stepsizes; and enjoys sublinear convergence rate in the general nonconvex case, and linear convergence case under strong convexity or under less stringent standard error bound conditions.The algorithmic framework we propose is very abstract and general and can be applied to different computing architectures (e.g., message-passing systems, cluster of computers, shared-memory environments), always converging under the same set of assumptions. </div><div></div><div>In the last Chapter we consider the case of distributed multi-agent systems. Indeed, in many practical applications the objective function has a favorable separable structure. In this case, we generalize our framework to take into consideration the presence of different agents, where each one of them knows only a portion of the overall function, which they want cooperatively to minimize. The result is the first fully decentralized asynchronous method for the setting described above. The proposed method achieve sublinear convergence rate in the general case, and linear convergence rate under standard error bound conditions.</div><div></div><div>Extensive simulation results on problems of practical interest (MRI reconstruction, LASSO, matrix completion) show that the proposed methods compare favorably to state-of-the art-schemes.</div>
Tor, Ali Hakan
01 February 2013
(has links) (PDF)
In this thesis, various numerical methods are developed to solve nonsmooth and in particular, nonconvex optimization problems. More speci
Distributed Optimization in Electric Power Systems: Partitioning, Communications, and SynchronizationGuo, Junyao 01 March 2018 (has links)
To integrate large volumes of renewables and use electricity more efficiently, many industrial trials are on-going around the world that aim to realize decentralized or hierarchical control of renewable and distributed energy resources, flexible loads and monitoring devices. As the cost and complexity involved in the centralized communications and control infrastructure may be prohibitive in controlling millions of these distributed energy resources and devices, distributed optimization methods are expected to become much more prevalent in the operation of future electric power systems, as they have the potential to address this challenge and can be applied to various applications such as optimal power ow, state estimation, voltage control, and many others. While many distributed optimization algorithms are developed mathematically, little effort has been reported so far on how these methods should actually be implemented in real-world large-scale systems. The challenges associated with this include identifying how to decompose the overall optimization problem, what communication infrastructures can support the information exchange among subproblems, and whether to coordinate the updates of the subproblems in a synchronous or asynchronous manner. This research is dedicated to developing mathematical tools to address these issues, particularly for solving the non-convex optimal power flow problem. As the first part of this thesis, we develop a partitioning method that defines the boundaries of regions when applying distributed algorithms to a power system. This partitioning method quantifies the computational couplings among the buses and groups the buses with large couplings into one region. Through numerical experiments, we show that the developed spectral partitioning approach is the key to achieving fast convergence of distributed optimization algorithms on large-scale systems. After the partitioning of the system is defined, one needs to determine whether the communications among neighboring regions are supported. Therefore, as the second part of this thesis, we propose models for centralized and distributed communications infrastructures and study the impact of communication delays on the efficiency of distributed optimization algorithms through network simulations. Our findings suggest that the centralized communications infrastructure can be prohibitive for distributed optimization and cost-effective migration paths to a more distributed communications infrastructure are necessary. As the sizes and complexities of subproblems and communication delays are generally heterogeneous, synchronous distributed algorithms can be inefficient as they require waiting for the slowest region in the system. Hence, as the third part of this thesis, we develop an asynchronous distributed optimization method and show its convergence for the considered optimal power flow problem. We further study the impact of parameter tuning, system partitioning and communication delays on the proposed asynchronous method and compare its practical performance with its synchronous counterpart. Simulation results indicate that the asynchronous approach can be more efficient with proper partitioning and parameter settings on large-scale systems. The outcome of this research provides important insights into how existing hardware and software solutions for Energy Management Systems in the power grid can be used or need to be extended for deploying distributed optimization methods, which establishes the interconnection between theoretical studies of distributed algorithms and their practical implementation. As the evolution towards a more distributed control architecture is already taking place in many utility networks, the approaches proposed in this thesis provide important tools and a methodology for adopting distributed optimization in power systems.
Multilevel optimization in infinity norm and associated stopping criteria / Optimisation multiniveaux en norme infinie et critères d’arrêt associésMouffe, Mélodie 10 February 2009 (has links)
Cette thèse se concentre sur l'étude d'un algorithme multi niveaux de régions de confiance en norme infinie, conçu pour la résolution de problèmes d'optimisation non linéaires de grande taille pouvant être soumis a des contraintes de bornes. L'étude est réalisée tant sur le plan théorique que numérique. L'algorithme RMTR8 que nous étudions ici a été élaboré a partir de l'algorithme présente par Gratton, Sartenaer et Toint (2008b), et modifie d'abord en remplaçant l'usage de la norme Euclidienne par une norme infinie, et ensuite en l'adaptant a la résolution de problèmes de minimisation soumis a des contraintes de bornes. Dans un premier temps, les spécificités du nouvel algorithme sont exposées et discutées. De plus, l'algorithme est démontré globalement convergent au sens de Conn, Gould et Toint (2000), c'est-a-dire convergent vers un minimum local au départ de tout point admissible. D'autre part, il est démontre que la propriété d'identification des contraintes actives des méthodes de régions de confiance basées sur l'utilisation d'un point de Cauchy peut être étendue a tout solveur interne respectant une décroissance suffisante. En conséquence, cette propriété d'identification est aussi respectée par une variante particulière du nouvel algorithme. Par la suite, nous étudions différents critères d'arrêt pour les algorithmes d'optimisation avec contraintes de bornes afin de déterminer le sens et les avantages de chacun, et ce pour pouvoir choisir aisément celui qui convient le mieux a certaines situations. En particulier, les critères d'arrêts sont analyses en termes d'erreur inverse (backward erreur), tant au sens classique du terme (avec l'usage d'une norme produit) que du point de vue de l'optimisation multicritères. Enfin, un algorithme pratique est mis en place, utilisant en particulier une technique similaire au lissage de Gauss-Seidel comme solveur interne. Des expérimentations numériques sont réalisées sur une version FORTRAN 95 de l'algorithme. Elles permettent d'une part de définir un panel de paramètres efficaces par défaut et, d'autre part, de comparer le nouvel algorithme a d'autres algorithmes classiques d'optimisation, comme la technique de raffinement de maillage ou la méthode du gradient conjugue, sur des problèmes avec et sans contraintes de bornes. Ces comparaisons numériques semblent donner l'avantage à l'algorithme multi niveaux, en particulier sur les cas peu non-linéaires, comportement attendu de la part d'un algorithme inspire des techniques multi grilles. En conclusion, l'algorithme de région de confiance multi niveaux présente dans cette thèse est une amélioration du précédent algorithme de cette classe d'une part par l'usage de la norme infinie et d'autre part grâce a son traitement de possibles contraintes de bornes. Il est analyse tant sur le plan de la convergence que de son comportement vis-à-vis des bornes, ou encore de la définition de son critère d'arrêt. Il montre en outre un comportement numérique prometteur. / This thesis concerns the study of a multilevel trust-region algorithm in infinity norm, designed for the solution of nonlinear optimization problems of high size, possibly submitted to bound constraints. The study looks at both theoretical and numerical sides. The multilevel algorithm RMTR8 that we study has been developed on the basis of the algorithm created by Gratton, Sartenaer and Toint (2008b), which was modified first by replacing the use of the Euclidean norm by the infinity norm and also by adapting it to solve bound-constrained problems. In a first part, the main features of the new algorithm are exposed and discussed. The algorithm is then proved globally convergent in the sense of Conn, Gould and Toint (2000), which means that it converges to a local minimum when starting from any feasible point. Moreover, it is shown that the active constraints identification property of the trust-region methods based on the use of a Cauchy step can be extended to any internal solver that satisfies a sufficient decrease property. As a consequence, this identification property also holds for a specific variant of our new algorithm. Later, we study several stopping criteria for nonlinear bound-constrained algorithms, in order to determine their meaning and their advantages from specific points of view, and such that we can choose easily the one that suits best specific situations. In particular, the stopping criteria are examined in terms of backward error analysis, which has to be understood both in the usual meaning (using a product norm) and in a multicriteria optimization framework. In the end, a practical algorithm is set on, that uses a Gauss-Seidel-like smoothing technique as an internal solver. Numerical tests are run on a FORTRAN 95 version of the algorithm in order to define a set of efficient default parameters for our method, as well as to compare the algorithm with other classical algorithms like the mesh refinement technique and the conjugate gradient method, on both unconstrained and bound-constrained problems. These comparisons seem to give the advantage to the designed multilevel algorithm, particularly on nearly quadratic problems, which is the behavior expected from an algorithm inspired by multigrid techniques. In conclusion, the multilevel trust-region algorithm presented in this thesis is an improvement of the previous algorithm of this kind because of the use of the infinity norm as well as because of its handling of bound constraints. Its convergence, its behavior concerning the bounds and the definition of its stopping criteria are studied. Moreover, it shows a promising numerical behavior.
STUDIES ON OPTIMIZATION PROBLEMS WITH POSITIVELY HOMOGENEOUS FUNCTIONS AND ASSOCIATED DUALITY RESULTS / 正斉次関数を含む最適化問題とその双対性に関する研究Yamanaka, Shota 24 September 2021 (has links)
京都大学 / 新制・課程博士 / 博士(情報学) / 甲第23546号 / 情博第776号 / 新制||情||132(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 山下 信雄, 教授 太田 快人, 教授 永持 仁 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM
The development of wireless sensor networks (WSNs) is making it possible to monitor our cities. Due to the small size of the sensor nodes, and their capabilities of transmitting data remotely, they can be deployed at locations that are not easy or impossible to access, such as the pipelines of water distribution networks (WDNs), which plays an important role in protecting environment and securing public health. The design of WSNs for WDNs faces major challenges. Generally, WSNs are resource-limited because most of the sensor nodes are battery powered. Thus, their resource allocation has to be carefully controlled. The thesis considers two prominent problems that occur when designing WSNs for WDNs: scheduling the sensing of the nodes of static WSNs, and sensor placement for mobile WSNs. These studies are reported in the thesis from three published or submitted papers. In the first paper, the scheduling of sleep/sensing for each sensor node is considered to maximize the whole WSNs lifetime while guaranteeing a monitoring performance constraint. The problem is transformed into an energy balancing problem, and solved by a dynamic programming based algorithm. It is proved that this algorithm finds one of the optimal solutions for the energy balancing problem. In the second paper, the question of how the energy balancing problem approximates the original scheduling problem is addressed. It is shown that even though these two problems are not equivalent, the gap of them is small enough. Thus, the proposed algorithm for the energy balancing problem can find a good approximation solution for the original scheduling problem. The second part of the thesis considers the use of mobile sensor nodes. Here, the limited resource is the number of available such mobile nodes. To maximize the monitoring coverage in terms of population, an optimization problem for determining the releasing locations for the mobile sensor nodes is formulated. An approximate solution algorithm based on submodular maximization is proposed and its performance is investigated. Beside WDNs, WSN applications for smart cities share a common characteristic: the area to monitor usually has a network structure. Therefore, the studies of this thesis can be potentially generalized for several IoT scenarios. / <p>QC 20160419</p>
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Metodo de direções interiores ao epígrafo - IED para otimização não diferenciável e não convexa via Dualidade Lagrangeana: estratégias para minimização da Lagrangeana aumentadaFranco, Hernando José Rocha 08 June 2018 (has links)
Submitted by Geandra Rodrigues (firstname.lastname@example.org) on 2018-07-12T12:23:47Z No. of bitstreams: 1 hernandojoserochafranco.pdf: 1674623 bytes, checksum: f6df7317dd6a8e94e51045dbf75e8241 (MD5) / Approved for entry into archive by Adriana Oliveira (email@example.com) on 2018-07-17T11:56:13Z (GMT) No. of bitstreams: 1 hernandojoserochafranco.pdf: 1674623 bytes, checksum: f6df7317dd6a8e94e51045dbf75e8241 (MD5) / Made available in DSpace on 2018-07-17T11:56:13Z (GMT). No. of bitstreams: 1 hernandojoserochafranco.pdf: 1674623 bytes, checksum: f6df7317dd6a8e94e51045dbf75e8241 (MD5) Previous issue date: 2018-06-08 / A teoria clássica de otimização presume a existência de certas condições, por exemplo, que as funções envolvidas em um problema desta natureza sejam pelo menos uma vez continuamente diferenciáveis. Entretanto, em muitas aplicações práticas que requerem o emprego de métodos de otimização, essa característica não se encontra presente. Problemas de otimização não diferenciáveis são considerados mais difíceis de lidar. Nesta classe, aqueles que envolvem funções não convexas são ainda mais complexos. O Interior Epigraph Directions (IED) é um método de otimização que se baseia na teoria da Dualidade Lagrangeana e se aplica à resolução de problemas não diferenciáveis, não convexos e com restrições. Neste estudo, apresentamos duas novas versões para o referido método a partir de implementações computacionais de outros algoritmos. A primeira versão, denominada IED+NFDNA, recebeu a incorporação de uma implementação do algoritmo Nonsmooth Feasible Direction Nonconvex Algorithm (NFDNA). Esta versão, ao ser aplicada em experimentos numéricos com problemas teste da literatura, apresentou desempenho satisfatório quando comparada ao IED original e a outros solvers de otimização. Com o objetivo de aperfeiçoar mais o método, reduzindo sua dependência de parâmetros iniciais e também do cálculo de subgradientes, uma segunda versão, IED+GA, foi desenvolvida com a utilização de algoritmos genéticos. Além da resolução de problemas teste, o IED-FGA obteve bons resultados quando aplicado a problemas de engenharia. / The classical theory of optimization assumes the existence of certain conditions, for example, that the functions involved in a problem of this nature are at least once continuously differentiable. However, in many practical applications that require the use of optimization methods, this characteristic is not present. Non-differentiable optimization problems are considered more difficult to deal with. In this class, those involving nonconvex functions are even more complex. Interior Epigraph Directions (IED) is an optimization method that is based on Lagrangean duality theory and applies to the resolution of non-differentiable, non-convex and constrained problems. In this study, we present two new versions for this method from computational implementations of other algorithms. The first version, called IED + NFDNA, received the incorporation of an implementation of the Nonsmooth Feasible Direction Nonconvex Algorithm (NFDNA) algorithm. This version, when applied in numerical experiments with problems in the literature, presented satisfactory performance when compared to the original IED and other optimization solvers. A second version, IED + GA, was developed with the use of genetic algorithms in order to further refine the method, reducing its dependence on initial parameters and also on the calculation of subgradients. In addition to solving test problems, IED + GA achieved good results when applied to engineering problems.
In economical and sustainable operation of cyber-physical systems, a number of entities need to often cooperate over a communication network to solve optimization problems. A challenging aspect in the design of robust distributed solution algorithms to these optimization problems is that as technology advances and the networks grow larger, the communication bandwidth used to coordinate the solution is limited. Moreover, even though most research has focused distributed convex optimization, in cyberphysical systems nonconvex problems are often encountered, e.g., localization in wireless sensor networks and optimal power flow in smart grids, the solution of which poses major technical difficulties. Motivated by these challenges this thesis investigates distributed optimization with emphasis on limited communication for both convex and nonconvex structured problems. In particular, the thesis consists of four articles as summarized below. The first two papers investigate the convergence of distributed gradient solution methods for the resource allocation optimization problem, where gradient information is communicated at every iteration, using limited communication. In particular, the first paper investigates how distributed dual descent methods can perform demand-response in power networks by using one-way communication. To achieve the one-way communication, the power supplier first broadcasts a coordination signal to the users and then updates the coordination signal by using physical measurements related to the aggregated power usage. Since the users do not communicate back to the supplier, but instead they only take a measurable action, it is essential that the algorithm remains primal feasible at every iteration to avoid blackouts. The paper demonstrates how such blackouts can be avoided by appropriately choosing the algorithm parameters. Moreover, the convergence rate of the algorithm is investigated. The second paper builds on the work of the first paper and considers more general resource allocation problem with multiple resources. In particular, a general class of quantized gradient methods are studied where the gradient direction is approximated by a finite quantization set. Necessary and sufficient conditions on the quantization set are provided to guarantee the ability of these methods to solve a large class of dual problems. A lower bound on the cardinality of the quantization set is provided, along with specific examples of minimal quantizations. Furthermore, convergence rate results are established that connect the fineness of the quantization and number of iterations needed to reach a predefined solution accuracy. The results provide a bound on the number of bits needed to achieve the desired accuracy of the optimal solution. The third paper investigates a particular nonconvex resource allocation problem, the Optimal Power Flow (OPF) problem, which is of central importance in the operation of power networks. An efficient novel method to address the general nonconvex OPF problem is investigated, which is based on the Alternating Direction Method of Multipliers (ADMM) combined with sequential convex approximations. The global OPF problem is decomposed into smaller problems associated to each bus of the network, the solutions of which are coordinated via a light communication protocol. Therefore, the proposed method is highly scalable. The convergence properties of the proposed algorithm are mathematically and numerically substantiated. The fourth paper builds on the third paper and investigates the convergence of distributed algorithms as in the third paper but for more general nonconvex optimization problems. In particular, two distributed solution methods, including ADMM, that combine the fast convergence properties of augmented Lagrangian-based methods with the separability properties of alternating optimization are investigated. The convergence properties of these methods are investigated and sufficient conditions under which the algorithms asymptotically reache the first order necessary conditions for optimality are established. Finally, the results are numerically illustrated on a nonconvex localization problem in wireless sensor networks. The results of this thesis advocate the promising convergence behaviour of some distributed optimization algorithms on nonconvex problems. Moreover, the results demonstrate the potential of solving convex distributed resource allocation problems using very limited communication bandwidth. Future work will consider how even more general convex and nonconvex problems can be solved using limited communication bandwidth and also study lower bounds on the bandwidth needed to solve general resource allocation optimization problems. / <p>QC 20160203</p>
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