Submitted by Cláudia Bueno (claudiamoura18@gmail.com) on 2015-10-29T19:04:41Z No. of bitstreams: 2 Tese - Tiberio Bittencourt de Oliveira Martins.pdf: 1155588 bytes, checksum: add1eac74c4397efc29678341b834448 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-11-03T14:25:04Z (GMT) No. of bitstreams: 2 Tese - Tiberio Bittencourt de Oliveira Martins.pdf: 1155588 bytes, checksum: add1eac74c4397efc29678341b834448 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-11-03T14:25:04Z (GMT). No. of bitstreams: 2 Tese - Tiberio Bittencourt de Oliveira Martins.pdf: 1155588 bytes, checksum: add1eac74c4397efc29678341b834448 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-06-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Apresentamos, nesta tese, uma an álise da convergência do m étodo de Newton inexato com tolerância de erro residual relativa e uma an alise semi-local de m etodos de Newton robustos exato e inexato, objetivando encontrar uma singularidade de um campo de vetores diferenci avel de nido em uma variedade Riemanniana completa, baseados no princ pio majorante a m invariante. Sob hip oteses locais e considerando uma fun ção majorante geral, a Q-convergância linear do m etodo de Newton inexato com uma tolerância de erro residual relativa xa e provada. Na ausência dos erros, a an alise apresentada reobtem o teorema local cl assico sobre o m etodo de Newton no contexto Riemanniano. Na an alise semi-local dos m etodos exato e inexato de Newton apresentada, a cl assica condi ção de Lipschitz tamb em e relaxada usando uma fun ção majorante geral, permitindo estabelecer existência e unicidade local da solu ção, uni cando previamente resultados pertencentes ao m etodo de Newton. A an alise enfatiza a robustez, a saber, e dada uma bola prescrita em torno do ponto inicial que satifaz as hip oteses de Kantorovich, garantindo a convergência do m etodo para qualquer ponto inicial nesta bola. Al em disso, limitantes que dependem da função majorante para a taxa de convergência Q-quadr atica do m étodo exato e para a taxa de convergência Q-linear para o m etodo inexato são obtidos. / A local convergence analysis with relative residual error tolerance of inexact Newton method and a semi-local analysis of a robust exact and inexact Newton methods are presented in this thesis, objecting to nd a singularity of a di erentiable vector eld de ned on a complete Riemannian manifold, based on a ne invariant majorant principle. Considering local assumptions and a general majorant function, the Q-linear convergence of inexact Newton method with a xed relative residual error tolerance is proved. In the absence of errors, the analysis presented retrieves the classical local theorem on Newton's method in Riemannian context. In the semi-local analysis of exact and inexact Newton methods presented, the classical Lipschitz condition is also relaxed by using a general majorant function, allowing to establish the existence and also local uniqueness of the solution, unifying previous results pertaining Newton's method. The analysis emphasizes robustness, being more speci c, is given a prescribed ball around the point satisfying Kantorovich's assumptions, ensuring convergence of the method for any starting point in this ball. Furthermore, the bounds depending on the majorant function for Q-quadratic convergence rate of the exact method and Q-linear convergence rate of the inexact method are obtained.

Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2017-09-21T21:10:08Z No. of bitstreams: 2 Tese - Yuri Rafael Leite Pereira - 2017.pdf: 2066899 bytes, checksum: e1bbe4df9a2a43e1074b83920a833ced (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-09-22T11:44:33Z (GMT) No. of bitstreams: 2 Tese - Yuri Rafael Leite Pereira - 2017.pdf: 2066899 bytes, checksum: e1bbe4df9a2a43e1074b83920a833ced (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-09-22T11:44:33Z (GMT). No. of bitstreams: 2 Tese - Yuri Rafael Leite Pereira - 2017.pdf: 2066899 bytes, checksum: e1bbe4df9a2a43e1074b83920a833ced (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-08-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we will analyze three types of method to solve vector optimization problems in different types of context. First, we will present the trust region method for multiobjective optimization in the Riemannian context, which retrieves the classical trust region method for minimizing scalar functions. Under mild assumptions, we will show that each accumulation point of the generated sequences by the method, if any, is Pareto critical. Next, the proximal point method for vector optimization and its inexact version will be extended from Euclidean space to the Riemannian context. Under suitable assumptions on the objective function, the well-definedness of the methods will be established. Besides, the convergence of any generated sequence, to a weak efficient point, will be obtained. The last method to be investigated is the Newton method to solve vector optimization problem with respect to variable ordering structure. Variable ordering structures are set-valued map with cone values that to each element associates an ordering. In this analyze we will prove the convergence of the sequence generated by the algorithm of Newton method and, moreover, we also will obtain the rate of convergence under variable ordering structures satisfying mild hypothesis. / Neste trabalho, analisaremos três tipos de métodos para resolver problemas de otimização vetorial em diferentes tipos contextos. Primeiro, apresentaremos o método da Região de Confiança para resolver problemas multiobjetivo no contexto Riemanniano, o qual recupera o método da Região de Confiança clássica para minimizar funções escalares. Sob determinadas suposições, mostraremos que cada ponto de acumulação das sequências geradas pelo método, se houver, é Pareto crítico. Em seguida, o método do ponto proximal para otimização vetorial e sua versão inexata serão estendidos do espaço Euclidiano para o contexto Riemanniano. Sob adequados pressupostos sobre a função objetiva, a boas definições dos métodos serão estabelecidos. Além disso, a convergência de qualquer sequência gerada, para um ponto fracamente eficiente, é obtida. O último método a ser investigado é o método de Newton para resolver o problema de otimização vetorial com respeito a estruturas de ordem variável. Estruturas de ordem variável são aplicações ponto-conjunto cujas imagens são cones que para cada elemento associa uma ordem. Nesta análise, provaremos a convergência da sequência gerada pelo algoritmo do método de Newton e, além disso, também obteremos a taxa de convergência sob estruturas de ordem variável satisfazendo adequadas hipóteses.

Submitted by JÚLIO HEBER SILVA (julioheber@yahoo.com.br) on 2017-03-22T20:23:25Z No. of bitstreams: 2 Tese - Gilson do Nascimento Silva - 2017.pdf: 2015008 bytes, checksum: e0148664ca46221978f71731aeabfa36 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-03-23T11:30:21Z (GMT) No. of bitstreams: 2 Tese - Gilson do Nascimento Silva - 2017.pdf: 2015008 bytes, checksum: e0148664ca46221978f71731aeabfa36 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-03-23T11:30:21Z (GMT). No. of bitstreams: 2 Tese - Gilson do Nascimento Silva - 2017.pdf: 2015008 bytes, checksum: e0148664ca46221978f71731aeabfa36 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-03-13 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We consider Newton’s method for solving a generalized equation of the form f(x) + F(x) 3 0, where f : Ω → Y is continuously differentiable, X and Y are Banach spaces, Ω ⊆ X is open and F : X ⇒ Y has nonempty closed graph. Assuming strong regularity of the equation and that the starting point satisfies Kantorovich’s conditions, we show that the method is quadratically convergent to a solution, which is unique in a suitable neighborhood of the starting point. In addition, a local convergence analysis of this method is presented. Moreover, using convex optimization techniques introduced by S. M. Robinson (Numer. Math., Vol. 19, 1972, pp. 341-347), we prove a robust convergence theorem for inexact Newton’s method for solving nonlinear inclusion problems in Banach space, i.e., when F(x) = −C and C is a closed convex set. Our analysis, which is based on Kantorovich’s majorant technique, enables us to obtain convergence results under Lipschitz, Smale’s and Nesterov-Nemirovskii’s self-concordant conditions. / N´os consideraremos o m´etodo de Newton para resolver uma equa¸c˜ao generalizada da forma f(x) + F(x) 3 0, onde f : Ω → Y ´e continuamente diferenci´avel, X e Y s˜ao espa¸cos de Banach, Ω ⊆ X ´e aberto e F : X ⇒ Y tem gr´afico fechado n˜ao-vazio. Supondo regularidade forte da equa¸c˜ao e que o ponto inicial satisfaz as hip´oteses de Kantorovich, mostraremos que o m´etodo ´e quadraticamente convergente para uma solu¸c˜ao, a qual ´e ´unica em uma vizinhan¸ca do ponto inicial. Uma an´alise de convergˆencia local deste m´etodo tamb´em ´e apresentada. Al´em disso, usando t´ecnicas de otimiza¸c˜ao convexa introduzida por S. M. Robinson (Numer. Math., Vol. 19, 1972, pp. 341-347), provaremos um robusto teorema de convergˆencia para o m´etodo de Newton inexato para resolver problemas de inclus˜ao n˜ao–linear em espa¸cos de Banach, i.e., quando F(x) = −C e C ´e um conjunto convexo fechado. Nossa an´alise, a qual ´e baseada na t´ecnica majorante de Kantorovich, nos permite obter resultados de convergˆencia sob as condi¸c˜oes Lipschitz, Smale e Nesterov-Nemirovskii auto-concordante.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-07-01T14:29:29Z No. of bitstreams: 1 paulaoliveiralagatta.pdf: 1917786 bytes, checksum: 627585584595873c205fcbcf5c79980f (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-07-13T16:01:23Z (GMT) No. of bitstreams: 1 paulaoliveiralagatta.pdf: 1917786 bytes, checksum: 627585584595873c205fcbcf5c79980f (MD5) / Made available in DSpace on 2016-07-13T16:01:23Z (GMT). No. of bitstreams: 1 paulaoliveiralagatta.pdf: 1917786 bytes, checksum: 627585584595873c205fcbcf5c79980f (MD5) Previous issue date: 2012-03-05 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho são propostas duas formulações de importantes ferramentas para análise de redes em regime permanente, onde são consideradas equações que descrevem o comportamento do controle primário e secundário de frequência em sistemas elétricos de potência. A primeira proposta é baseada em uma formulação do problema de fluxo de potência convencional e a segunda uma formulação do fluxo de potência ótimo. A formulação de fluxo de potência proposta é desenvolvida a partir de uma metodologia genérica de representação de dispositivos de controle. Esta metodologia consiste em incorporar as equações que modelam dispositivos de controle ao problema básico de fluxo de potência em coordenadas polares, formando um sistema de equações de ordem (2nb+nc). O fluxo de potência desenvolvido é capaz de estimar os desvios de frequência do sistema devido a uma perturbação da carga. Por outro lado, o fluxo de potência ótimo proposto é capaz de identificar montantes e locais de corte carga, de forma a manter a frequência do sistema em uma faixa aceitável de operação. A formulação proposta de FPO consiste em incluir no problema equações de igualdade e desigualdade associadas com o controle primário de frequência e geração de potência ativa. Os desenvolvimentos propostos para o fluxo de potência convencional foram implementados no ambiente MatLab®. Para solução do fluxo de potência ótimo utilizou-se um pacote comercial de otimização, denominado LINGO®. A avaliação do fluxo de potência e fluxo de potência ótimo propostos é feita através do estudo de sistemas tutoriais e do sistema New England. A validação da análise de desvios de frequência é feita através da utilização do programa ANATEM, desenvolvido pelo CEPEL. Os resultados obtidos mostram as vantagens da utilização das formulações propostas. / This work proposes a new formulation for both the conventional power flow and the optimal power flow formulation, in which the steady-state equations describing the primary and secondary frequency control in electrical power systems are included. The proposed power flow formulation is based on a flexible methodology for the representation of control devices. Such methodology incorporates equations that model control devices into the basic power flow formulation in polar coordinates, generating an augmented system of equations having order (2nb + nc). The developed power flow is able to estimate the system frequency deviation due to a load disturbance. On other hand, the proposed optimum power flow formulation is able to identify the minimum load shedding necessary to maintain the system frequency in an acceptable range of operation. The proposed OPF formulation includes additional equality and inequality constraints to represent the steady state primary frequency control as a function of the active power generation. The proposed development for the conventional power flow was made using the MATLAB® environment. The optimal power flow solution used a commercial optimization package called LINGO®. The evaluation of the proposed power flow and optimal power flow formulations were made through the study of small test systems and the New England test system. Validations of the frequency deviation analysis were made using the program ANATEM, developed by CEPEL. The results obtained show the advantages of using the proposed formulations.

This thesis is broadly concerned with online-optimizing anti-windup control. These are control structures that implement some online-optimization routines to compensate for the windup effects in constrained control systems. The first part of this thesis examines a general framework for analyzing robust preservation in anti-windup control systems. This framework - the robust Kalman conjecture - is defined for the robust Lur’e problem. This part of the thesis verifies this conjecture for first-order plants perturbed by various norm-bounded unstructured uncertainties. Integral quadratic constraint theory is exploited to classify the appropriate stability multipliers required for verification in these cases. The remaining part of the thesis focusses on accelerated gradient methods. In particular, tight complexity-certificates can be obtained for the Nesterov gradient method, which makes it attractive for implementation of online-optimizing anti-windup control. This part of the thesis presents a proposed algorithm that extends the classical Nesterov gradient method by using available secant information. Numerical results demonstrating the efficiency of the proposed algorithm are analysed with the aid of performance profiles. As the objective function becomes more ill-conditioned, the proposed algorithm becomes significantly more efficient than the classical Nesterov gradient method. The improved performance bodes well for online-optimization anti-windup control since ill-conditioning is common place in constrained control systems. In addition, this thesis explores another subcategory of accelerated gradient methods known as Barzilai-Borwein gradient methods. Here, two algorithms that modify the Barzilai-Borwein gradient method are proposed. Global convergence of the proposed algorithms for all convex functions is established by using discrete Lyapunov theorems.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2015-12-16T16:23:08Z No. of bitstreams: 1 jhonatannascimentodacosta.pdf: 4337030 bytes, checksum: 25cf98ccb23a326195b9a1a90c7d43df (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2015-12-17T11:06:06Z (GMT) No. of bitstreams: 1 jhonatannascimentodacosta.pdf: 4337030 bytes, checksum: 25cf98ccb23a326195b9a1a90c7d43df (MD5) / Made available in DSpace on 2015-12-17T11:06:06Z (GMT). No. of bitstreams: 1 jhonatannascimentodacosta.pdf: 4337030 bytes, checksum: 25cf98ccb23a326195b9a1a90c7d43df (MD5) Previous issue date: 2015-02-27 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho são propostos dois índices eficientes baseados em Equivalentes de Thévenin para avaliação da segurança de tensão de Sistemas Elétricos de Potência de grande porte. Estes índices são denominados de Índice de Estabilidade de Tensão e Índice de Perda de Controle de Tensão e baseiam-se na característica de máxima transferência de potência de circuitos elétricos lineares. Neste sentido, propõe-se uma nova metodologia para a estimação da impedância de Thévenin baseada na técnica de Análise de Sensibilidade da matriz Jacobiana do problema de Fluxo de Potência. O Índice de Estabilidade de Tensão proposto pode ser calculado para todas as barras do sistema em um dado ponto de operação, fornecendo uma estimativa rápida do ponto de vista computacional da Margem de Carregamento e uma indicação das barras críticas do sistema. Por outro lado, o Índice de Perda de Controle de Tensão é calculado somente para as barras do tipo PV, fornecendo uma indicação dos geradores críticos para o controle de tensão da região em análise. Os índices propostos são avaliados através do estudo de sistemas tutoriais, de sistemas de médio porte e de um sistema de grande porte baseado no Sistema Interligado Nacional brasileiro. Sempre que possível procura-se validar os resultados obtidos através de comparações com as técnicas do vetor tangente do Método da Continuação e de menor Margem de Potência Reativa das Curvas V-Q, que são técnicas já consagradas para análise de estabilidade de tensão de Sistemas Elétricos de Potência. / In this work are proposed two efficient indexes based on Thévenin equivalent for assessment of voltage safety of large Electric Power Systems. These indexes are called Voltage Stability Index and Voltage Control Loss Index and are based on the characteristic of maximum power transfer of linear electrical circuits. In this sense, we propose a new methodology to estimate the Thévenin impedance based on the Sensitivity Analysis technique of the Jacobian matrix of power flow problem. The Voltage Stability Index proposed can be calculated for all system buses at a given operating point, providing a quick estimate of the computational point of view of the Loading Margin and an indication of the critical buses of the system. On the other hand, the Voltage Control Loss Index is calculated only for the PV type buses and provides an indication of the critical generators for control of voltage of the region in analisys. The proposed indexes are evaluated by studying tutorials systems, medium size systems and a large system based on the brazilian National Interconnected System. Whenever possible looking up validate the results obtained through comparisons with the techniques of the tangent vector of the Continuation Method and of the smaller Reactive Power Margin of the V-Q curves, which already are established techniques of voltage stability analysis of Electric Power Systems.

A class of Lagrange-Newton-SQP methods is investigated for optimal control problems governed by semilinear parabolic initial- boundary value problems. Distributed and boundary controls are given, restricted by pointwise upper and lower bounds. The convergence of the method is discussed in appropriate Banach spaces. Based on a weak second order sufficient optimality condition for the reference solution, local quadratic convergence is proved. The proof is based on the theory of Newton methods for generalized equations in Banach spaces.

We present a new method for the computation of low rank approximations to the solution of large, sparse, stable Lyapunov equations. It is based on a generalization of the classical Smith method and profits by the usual low rank property of the right hand side matrix. The requirements of the method are moderate with respect to both computational cost and memory. Hence, it provides a possibility to tackle large scale control problems. Besides the efficient solution of the matrix equation itself, a thorough integration of the method into several control algorithms can improve their performance to a high degree. This is demonstrated for algorithms for model reduction and optimal control. Furthermore, we propose a heuristic for determining a set of suboptimal ADI shift parameters. This heuristic, which is based on a pair of Arnoldi processes, does not require any a priori knowledge on the spectrum of the coefficient matrix of the Lyapunov equation. Numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the ADI parameters.

This thesis addresses optimal control problems with elasticity for large deformations. A hyperelastic model with a polyconvex energy density is employed to describe the elastic behavior of a body. The two approaches to derive the nonlinear partial differential equation, a balance of forces and an energy minimization, are compared. Besides the conventional volume and boundary loads, two novel internal loads are presented. Furthermore, curvilinear coordinates and a hierarchical plate model can be incorporated into the formulation of the elastic forward problem. The forward problem can be solved with Newton\\\'s method, though a globalization technique should be used to avoid divergence of Newton\\\'s method. The repeated solution of the Newton system is done by a CG or MinRes method with a multigrid V-cycle as a preconditioner. The optimal control problem consists of the displacement (as the state) and a load (as the control). Besides the standard tracking-type objective, alternative objective functionals are presented for problems where a reasonable desired state cannot be provided. Two methods are proposed to solve the optimal control problem: an all-at-once approach by a Lagrange-Newton method and a reduced formulation by a quasi-Newton method with an inverse limited-memory BFGS update. The algorithms for the solution of the forward problem and the optimal control problem are implemented in the finite-element software FEniCS, with the geometrical multigrid extension FMG. Numerical experiments are performed to demonstrate the mesh independence of the algorithms and both optimization methods.

In this thesis we present research on mathematical properties of methods for solv- ing symmetric systems of linear equations that arise in various optimization problem formulations and in methods for solving such problems. In the first and third paper (Paper A and Paper C), we consider the connection be- tween the method of conjugate gradients and quasi-Newton methods on strictly convex quadratic optimization problems or equivalently on a symmetric system of linear equa- tions with a positive definite matrix. We state conditions on the quasi-Newton matrix and the update matrix such that the search directions generated by the corresponding quasi-Newton method and the method of conjugate gradients respectively are parallel. In paper A, we derive such conditions on the update matrix based on a sufficient condition to obtain mutually conjugate search directions. These conditions are shown to be equivalent to the one-parameter Broyden family. Further, we derive a one-to-one correspondence between the Broyden parameter and the scaling between the search directions from the method of conjugate gradients and a quasi-Newton method em- ploying some well-defined update scheme in the one-parameter Broyden family. In paper C, we give necessary and sufficient conditions on the quasi-Newton ma- trix and on the update matrix such that equivalence with the method of conjugate gra- dients hold for the corresponding quasi-Newton method. We show that the set of quasi- Newton schemes admitted by these necessary and sufficient conditions is strictly larger than the one-parameter Broyden family. In addition, we show that this set of quasi- Newton schemes includes an infinite number of symmetric rank-one update schemes. In the second paper (Paper B), we utilize an unnormalized Krylov subspace frame- work for solving symmetric systems of linear equations. These systems may be incom- patible and the matrix may be indefinite/singular. Such systems of symmetric linear equations arise in constrained optimization. In the case of an incompatible symmetric system of linear equations we give a certificate of incompatibility based on a projection on the null space of the symmetric matrix and characterize a minimum-residual solu- tion. Further we derive a minimum-residual method, give explicit recursions for the minimum-residual iterates and characterize a minimum-residual solution of minimum Euclidean norm. / I denna avhandling betraktar vi matematiska egenskaper hos metoder för att lösa symmetriska linjära ekvationssystem som uppkommer i formuleringar och metoder för en mängd olika optimeringsproblem. I första och tredje artikeln (Paper A och Paper C), undersöks kopplingen mellan konjugerade gradientmetoden och kvasi-Newtonmetoder när dessa appliceras på strikt konvexa kvadratiska optimeringsproblem utan bivillkor eller ekvivalent på ett symmet- risk linjärt ekvationssystem med en positivt definit symmetrisk matris. Vi ställer upp villkor på kvasi-Newtonmatrisen och uppdateringsmatrisen så att sökriktningen som fås från motsvarande kvasi-Newtonmetod blir parallell med den sökriktning som fås från konjugerade gradientmetoden. I den första artikeln (Paper A), härleds villkor på uppdateringsmatrisen baserade på ett tillräckligt villkor för att få ömsesidigt konjugerade sökriktningar. Dessa villkor på kvasi-Newtonmetoden visas vara ekvivalenta med att uppdateringsstrategin tillhör Broydens enparameterfamilj. Vi tar också fram en ett-till-ett överensstämmelse mellan Broydenparametern och skalningen mellan sökriktningarna från konjugerade gradient- metoden och en kvasi-Newtonmetod som använder någon väldefinierad uppdaterings- strategi från Broydens enparameterfamilj. I den tredje artikeln (Paper C), ger vi tillräckliga och nödvändiga villkor på en kvasi-Newtonmetod så att nämnda ekvivalens med konjugerade gradientmetoden er- hålls. Mängden kvasi-Newtonstrategier som uppfyller dessa villkor är strikt större än Broydens enparameterfamilj. Vi visar också att denna mängd kvasi-Newtonstrategier innehåller ett oändligt antal uppdateringsstrategier där uppdateringsmatrisen är en sym- metrisk matris av rang ett. I den andra artikeln (Paper B), används ett ramverk för icke-normaliserade Krylov- underrumsmetoder för att lösa symmetriska linjära ekvationssystem. Dessa ekvations- system kan sakna lösning och matrisen kan vara indefinit/singulär. Denna typ av sym- metriska linjära ekvationssystem uppkommer i en mängd formuleringar och metoder för optimeringsproblem med bivillkor. I fallet då det symmetriska linjära ekvations- systemet saknar lösning ger vi ett certifikat för detta baserat på en projektion på noll- rummet för den symmetriska matrisen och karaktäriserar en minimum-residuallösning. Vi härleder även en minimum-residualmetod i detta ramverk samt ger explicita rekur- sionsformler för denna metod. I fallet då det symmetriska linjära ekvationssystemet saknar lösning så karaktäriserar vi en minimum-residuallösning av minsta euklidiska norm. / <p>QC 20150519</p>