Orientador: Francisco de Assis Magalhães Gomes Neto / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-24T00:24:19Z (GMT). No. of bitstreams: 1 Siqueira_AbelSoares_D.pdf: 1465105 bytes, checksum: 2cba750df9607e9cb37b5799b157c850 (MD5) Previous issue date: 2013 / Resumo: Uma maneira de resolver problemas gerais de programação não linear é utilizar estratégias de passos compostos. Essas estratégias normalmente combinam um passo tangente às restrições e um passo normal, alternando entre a diminuição da função objetivo e da norma da infactibilidade. Esse tipo de método exige o controle dos passos ou dos iterandos, para que não se perca o progresso de um vii passo no outro. Apresentaremos uma extensão do método de Controle Dinâmico da Infactibilidade, que utiliza uma estratégia de controle de passos chamado de Cilindros de Confiança. Esse método foi desenvolvido para problemas com restrições apenas de igualdade, e nossa extensão lida com restrições gerais. Mostraremos testes numéricos comparando nosso método com um método do mesmo tipo / Abstract: One way to solve general nonlinear programming problems is the composite-step strategies. These strategies usually combine a step tangent to the constraints and a normal step, alternating between reducing the objective function value and the norm of the infeasibility. This kind of method requires the control of the steps or the iterates, in order to prevent one step from destroying the progress of another. We will present an extension of the Dynamic Control of Infeasibility method, which utilizes a strategy to control the steps known as Trust Cylinders. This method was originally designed for problems with equality contraints only, and our extension will handle general constraints. We'll show numerical experiments comparing our method with another composite-step method / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

Orientador: Maria Aparecida Diniz Ehrhardt / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação / Made available in DSpace on 2018-08-22T06:10:21Z (GMT). No. of bitstreams: 1 Ferreira_DeiseGoncalves_M.pdf: 2631020 bytes, checksum: 45eb84901394375843735b1fdef599ad (MD5) Previous issue date: 2013 / Resumo: Neste trabalho voltamos nossa atenção para métodos de otimização que não fazem uso de derivadas. Dentre esses, estamos interessadas em um método de busca padrão para minimização de funções com restrições lineares. Abordamos um algoritmo proposto por Lewis e Torczon, cuja ideia geral é que o padrão deve conter direções de busca ao longo das quais iterações factíveis sejam determinadas. O algoritmo possui resultados de convergência global. Realizamos sua implementação computacional, e propomos novas estratégias de busca e atualização do tamanho do passo, além de um novo padrão de direções de busca. Realizamos testes numéricos, de modo a analisar o desempenho das estratégias propostas e comparar o desempenho do padrão de direções que introduzimos com o proposto por Lewis e Torczon / Abstract: In this work, our interest lies on derivative-free optimization methods. Among these, our aim is to study a pattern search method for linearly constrained minimization. We studied an algorithm proposed by Lewis and Torsion, whose general idea is that the pattern must contain search directions in which feasible iterations must be computed. The algorithm has global convergence results. We accomplished its computational implementation and we propose new strategies of search and updating rule for the step-length control parameter. We also propose a new pattern of search directions. We accomplished numerical experiments in order to analyze the performance of our proposals and also to compare the performance of our pattern with the one proposed by Lewis and Torczon / Mestrado / Matematica Aplicada / Mestra em Matemática Aplicada

Problemas de otimização em caixas são de grande importância, não só por surgirem naturalmente na formulação de problemas da vida prática, mas também por aparecerem como subproblemas de métodos de penalização ou do tipo Lagrangiano Aumentado para resolução de problemas de programação não-linear. O objetivo do trabalho é estudar um algoritmo de restrições ativas para problemas de otimização em caixas recentemente apresentado chamado ASA e compará-lo à versão mais recente de GENCAN, que é também um método de restrições ativas. Para tanto, foi elaborada uma metodologia de testes robusta e minuciosa, que se propõe a remediar vários dos aspectos comumente criticados em trabalhos anteriores. Com isso, puderam ser extraídas conclusões que levaram à melhoria de GENCAN, conforme ficou posteriormente comprovado por meio da metodologia aqui introduzida. / Box-constrained optimization problems are of great importance not only for naturally arising in several real-life problems formulation, but also for their occurrence as sub-problems in both penalty and Augmented Lagrangian methods for solving nonlinear programming problems. This work aimed at studying a recently introduced active-set method for box-constrained optimization called ASA and comparing it to the latest version of GENCAN, which is also an active-set method. For that purpose, we designed a robust and thorough testing methodology intended to remedy many of the widely criticized aspects of prior works. Thereby, we could draw conclusions leading to GENCAN\'s further development, as it later became evident by means of the same methodology herein proposed.

abstract: Breeding seeds to include desirable traits (increased yield, drought/temperature resistance, etc.) is a growing and important method of establishing food security. However, besides breeder intuition, few decision-making tools exist that can provide the breeders with credible evidence to make decisions on which seeds to progress to further stages of development. This thesis attempts to create a chance-constrained knapsack optimization model, which the breeder can use to make better decisions about seed progression and help reduce the levels of risk in their selections. The model’s objective is to select seed varieties out of a larger pool of varieties and maximize the average yield of the “knapsack” based on meeting some risk criteria. Two models are created for different cases. First is the risk reduction model which seeks to reduce the risk of getting a bad yield but still maximize the total yield. The second model considers the possibility of adverse environmental effects and seeks to mitigate the negative effects it could have on the total yield. In practice, breeders can use these models to better quantify uncertainty in selecting seed varieties / Dissertation/Thesis / Masters Thesis Industrial Engineering 2019

<div>Stochastic programs are standard models for decision-making under uncertainty and have been extensively studied in the operations research literature. In general, stochastic programming involves minimizing an expected cost function, where the expectation is with respect to fully specified stochastic models that quantify the aleatoric or `inherent' uncertainty in the decision-making problem. In practice, however, the stochastic models are unknown but can be estimated from data, introducing an additional epistemic uncertainty into the decision-making problem. The Bayesian framework provides a coherent way to quantify the epistemic uncertainty through the posterior distribution by combining prior beliefs of the decision-makers with the observed data. Bayesian methods have been used for data-driven decision-making in various applications such as inventory management, portfolio design, machine learning, optimal scheduling, and staffing, etc.</div><div> </div><div>Bayesian methods are challenging to implement, mainly due to the fact that the posterior is computationally intractable, necessitating the computation of approximate posteriors. Broadly speaking, there are two methods in the literature implementing approximate posterior inference. First are sampling-based methods such as Markov Chain Monte Carlo. Sampling-based methods are theoretically well understood, but they suffer from various issues like high variance, poor scalability to high-dimensional problems, and have complex diagnostics. Consequently, we propose to use optimization-based methods collectively known as variational inference (VI) that use information projections to compute an approximation to the posterior. Empirical studies have shown that VI methods are computationally faster and easily scalable to higher-dimensional problems and large datasets. However, the theoretical guarantees of these methods are not well understood. Moreover, VI methods are empirically and theoretically less explored in the decision-theoretic setting.</div><div><br></div><div> In this thesis, we first propose a novel VI framework for risk-sensitive data-driven decision-making, which we call risk-sensitive variational Bayes (RSVB). In RSVB, we jointly compute a risk-sensitive approximation to the `true' posterior and the optimal decision by solving a minimax optimization problem. The RSVB framework includes the naive approach of first computing a VI approximation to the true posterior and then using it in place of the true posterior for decision-making. We show that the RSVB approximate posterior and the corresponding optimal value and decision rules are asymptotically consistent, and we also compute their rate of convergence. We illustrate our theoretical findings in both parametric as well as nonparametric setting with the help of three examples: the single and multi-product newsvendor model and Gaussian process classification. Second, we present the Bayesian joint chance-constrained stochastic program (BJCCP) for modeling decision-making problems with epistemically uncertain constraints. We discover that using VI methods for posterior approximation can ensure the convexity of the feasible set in (BJCCP) unlike any sampling-based methods and thus propose a VI approximation for (BJCCP). We also show that the optimal value computed using the VI approximation of (BJCCP) are statistically consistent. Moreover, we derive the rate of convergence of the optimal value and compute the rate at which a VI approximate solution of (BJCCP) is feasible under the true constraints. We demonstrate the utility of our approach on an optimal staffing problem for an M/M/c queue. Finally, this thesis also contributes to the growing literature in understanding statistical performance of VI methods. In particular, we establish the frequentist consistency of an approximate posterior computed using a well known VI method that computes an approximation to the posterior distribution by minimizing the Renyi divergence from the ‘true’ posterior.</div>

Cardinality constrained optimization problems (CCOP) are a new class of optimization problems with many applications. In this thesis, we propose a framework called mathematical programs with disjunctive subspaces constraints (MPDSC), a special case of mathematical programs with disjunctive constraints (MPDC), to investigate CCOP. Our method is different from the relaxed complementarity-type reformulation in the literature. The first contribution of this thesis is that we study various stationarity conditions for MPDSC, and then apply them to CCOP. In particular, we recover disjunctive-type strong (S-) stationarity and Mordukhovich (M-) stationarity for CCOP, and then reveal the relationship between them and those from the relaxed complementarity-type reformulation. The second contribution of this thesis is that we obtain some new results for MPDSC, which do not hold for MPDC in general. We show that many constraint qualifications like the relaxed constant positive linear dependence (RCPLD) coincide with their piecewise versions for MPDSC. Based on such result, we prove that RCPLD implies error bounds for MPDSC. These two results also hold for CCOP. All of these disjunctive-type constraint qualifications for CCOP derived from MPDSC are weaker than those from the relaxed complementarity-type reformulation in some sense. / Graduate

The Unit Commitment Problem (UCP) poses a significant challenge in optimizing powergeneration schedules within complex and dynamic energy systems. This study explores theapplication of Genetic Algorithms (GAs) as a promising approach to address UCP, their ability tonavigate complex solution spaces and adapt to changing operational conditions. The work provides a broad exploration of their effectiveness, challenges, and future prospects. The objective of UCP is to efficiently optimize power generation schedules within complex energy systems, seeking cost-effective and reliable solutions while accommodating various operational constraints. Various encoding techniques and GA operations are implemented and evaluated incomparison to the solutions obtained from a commercial Mixed-Integer Linear Programming (MILP) solver. The key findings point to the potential for achieving high quality solutions and robustness in the application of these techniques. However, it is important to acknowledge and address challenges such as encoding complexity, extensive computation times, the risk of premature convergence, and the complications of handling complex constraints that continue to exist in this domain. The future scope lies in hybrid approaches, scalability enhancement and incorporation of multi-objective optimization, offering unrealized potential for the efficient andsustainable operation of modern energy systems.

Due to modularity and easy scalability, modular multilevel converters (MMCs) are deemed the most suitable for high-voltage and medium-voltage power conversion applications. However, large module capacitors are usually required in MMCs to store large circulating power of line-frequency and its harmonics that flow through the capacitors. Even though several methods for minimizing the circulating power have been proposed in the literature, there is still the need for a systematic and simplified approach of addressing these control strategies and evaluating their efficacy. Moreover, the generally accepted feedback control architecture for the MMC is complicated, derived through a rigorous mathematical analysis, and therefore, not easy to intuitively comprehend. Recently, a method of modeling of the MMC based on state-plane analysis and coordinate transformation, is proposed in the literature. Based on the state-plane analysis, two kinds of circulating power in the MMC are identified that are orthogonal to each other. This means these two circulating power can be controlled individually without affecting each other. To control these circulating power, in the literature, a decoupled equivalent circuit model is developed through the coordinate transformation which clearly suggests a means for minimizing these circulating power. Further extending this work, in this thesis, the existing control concepts for reducing the circulating power are unveiled in a systematic and simplified manner utilizing the decoupled equivalent circuit model. A graphical visualization of circulating power using the state-planes is provided for each control strategy to readily compare its efficacy. Moreover, the generally accepted control architecture of the MMC is presented in an intuitive and simplified way using the decoupled circuit model. The important physics related to control implementation, originally hidden behind the complicated mathematics, is explained in detail. / Master of Science / A power converter is an electrical device that converts electrical energy from one form to another in order to be compatible with the load demand. A typical power converter consists of semiconductor switches, inductor, capacitor etc. These power converters are required in a wide range of applications: automotive and traction, motor drives, renewable energy conversion, energy storage, aircraft, power generation, transmission, and distribution, to name a few. Many of these applications are continuously increasing their power capacity to handle the escalating demands of energy that exist due to rising population numbers, industrialization, urbanization etc. Consequently, it has been a responsibility of power electronics engineers and researchers to develop power converters that can handle high voltages and high currents. Multilevel power converters have been the key-enabling developments that can withstand high-voltages while using traditional low-voltage semiconductor switches. Several multilevel converters such as the neutral point clamped converter, flying capacitor converter, cascaded H-bridge converter, modular multilevel converter (MMC) etc. have been developed and commercialized in the last two decades. Among them, the MMC is a widely accepted topology for medium- and high-voltage power conversion applications. In an MMC, several modules are stacked together in series, and each module consists of semiconductor switches and a capacitor. The series connection of the modules enables the MMC to handle high-voltage power conversion using low-voltage traditional semiconductor switches. The voltage rating of an MMC can be easily scaled-up by simply increasing the number of modules in each arm. Moreover, since several identical modules are connected in each arm, the structure of the MMC is highly modular which helps greatly in manufacturing and design. Nonetheless, in MMCs, generally large circulating power flow to the capacitor in each module, which leads to significant voltage ripples. To suppress these voltage ripples, a large capacitor is required in each module, leading to large size and weight of the converter. In the literature, several control strategies have been proposed to minimize the circulating power. However, there is still the need for a systematic and simplified approach of addressing these control strategies and evaluating their efficacy. Moreover, the generally accepted feedback control architecture for the MMC is complicated, derived through a rigorous mathematical analysis, and therefore, not easy to intuitively comprehend. Recently, a decoupled equivalent circuit model has been developed in the literature. This model clearly explains the process of power flow in the MMC between input and output and the nature of the circulating power. The equivalent circuit model provides the circulating power, that are orthogonal to each other, meaning they can be controlled individually without affecting each other. Moreover, the equivalent circuit model clearly suggests a means for minimize the circulating power by providing two "ideal" control laws. Further extending this work, in this thesis, the existing control concepts for reducing the circulating power are unveiled in a systematic and simplified manner utilizing the decoupled equivalent circuit model. Moreover, the generally accepted control architecture of the MMC is presented in an intuitive and simplified way via the decoupled circuit model. The important physics related to control implementation, originally hidden behind the complicated mathematics, is explained in detail.

Constrained optimization is pivotal in deep learning for enforcing predefined operational, ethical, and regulatory constraints in models. This thesis explores iterative gradient-based methods for solving constrained optimization. These methods are preferred for their adaptability in the non-convex, stochastic, and high-dimensional learning landscape of neural networks. Lagrangian methods, which integrate constraints directly into the objective function using Lagrange multipliers, are particularly effective. This formulation facilitates a unified approach to handle both the model's performance and its adherence to constraints simultaneously. However, the min-max problems inherent in Lagrangian frameworks often suffer from unstable oscillatory dynamics. Solving them using traditional gradient descent-ascent methods potentially leads to suboptimal convergence and prolonged training times. To mitigate these issues, this thesis introduces the νPI algorithm, a novel algorithm that integrates Proportional-Integral (PI) control theory into the optimization of Lagrange multipliers. The νPI algorithm is designed to stabilize the optimization process by adjusting the updates to the Lagrange multipliers based on the current and past constraint violations, effectively dampening the oscillations typically observed with gradient descent-ascent methods. Experimental validations across various deep learning applications demonstrate that νPI not only stabilizes the learning process but also accelerates convergence to optimal or near-optimal solutions, outperforming traditional methods in terms of reliability and speed. Through its application, νPI proves to be a robust tool in managing the dynamics of constrained optimization, enhancing the capability of deep learning models to meet stringent external constraints. / L’optimisation contrainte est essentielle dans l’apprentissage profond pour faire respecter les contraintes opérationnelles, éthiques et réglementaires prédéfinies dans les modèles. Cette thèse explore les méthodes itératives basées sur le gradient pour résoudre l’optimisation contrainte. Ces méthodes sont privilégiées pour leur adaptabilité dans le paysage d’apprentissage non convexe, stochastique et de haute dimension des réseaux neuronaux. Les méthodes lagrangiennes, qui intègrent directement les contraintes dans la fonction objectif à l’aide de multiplicateurs de Lagrange, sont particulièrement efficaces. Cette formulation facilite une approche unifiée pour gérer simultanément la performance du modèle et son adhérence aux contraintes. Cependant, les problèmes min-max inhérents aux cadres lagrangiens souffrent souvent de dynamiques oscillatoires instables. Leur résolution à l’aide des méthodes traditionnelles de descente de gradient-ascent peut potentiellement conduire à une convergence sous-optimale et à des temps de formation prolongés. Pour atténuer ces problèmes, cette thèse introduit l’algorithme νPI, un nouvel algorithme qui intègre la théorie de contrôle Proportionnel-Intégral (PI) dans l’optimisation des multiplicateurs de Lagrange. L’algorithme νPI est conçu pour stabiliser le processus d’optimisation en ajustant les mises à jour des multiplicateurs de Lagrange en fonction des violations de contraintes actuelles et passées, atténuant efficacement les oscillations typiquement observées avec les méthodes de descente de gradient-ascent. Les validations expérimentales à travers diverses applications d’apprentissage profond démontrent que νPI stabilise non seulement le processus d’apprentissage mais accélère également la convergence vers des solutions optimales ou quasi-optimales, surpassant les méthodes traditionnelles en termes de fiabilité et de vitesse. Grâce à son application, νPI se révèle être un outil robuste pour gérer les dynamiques de l’optimisation contrainte, améliorant la capacité des modèles d’apprentissage profond à répondre à des contraintes externes strictes.

In this thesis, we develop preconditioned iterative methods for the solution of matrix systems arising from PDE-constrained optimization problems. In order to do this, we exploit saddle point theory, as this is the form of the matrix systems we wish to solve. We utilize well-known results on saddle point systems to motivate preconditioners based on effective approximations of the (1,1)-block and Schur complement of the matrices involved. These preconditioners are used in conjunction with suitable iterative solvers, which include MINRES, non-standard Conjugate Gradients, GMRES and BiCG. The solvers we use are selected based on the particular problem and preconditioning strategy employed. We consider the numerical solution of a range of PDE-constrained optimization problems, namely the distributed control, Neumann boundary control and subdomain control of Poisson's equation, convection-diffusion control, Stokes and Navier-Stokes control, the optimal control of the heat equation, and the optimal control of reaction-diffusion problems arising in chemical processes. Each of these problems has a special structure which we make use of when developing our preconditioners, and specific techniques and approximations are required for each problem. In each case, we motivate and derive our preconditioners, obtain eigenvalue bounds for the preconditioners where relevant, and demonstrate the effectiveness of our strategies through numerical experiments. The goal throughout this work is for our iterative solvers to be feasible and reliable, but also robust with respect to the parameters involved in the problems we consider.