O método do Gradiente Espectral, introduzido por Barzilai e Borwein e analisado por Raydan, para minimização irrestrita, é um método simples cujo desempenho é comparável ao de métodos tradicionais como, por exemplo, gradientes conjugados. Desde a introdução do método, assim como da sua extensão para minimização em conjuntos convexos, foram introduzidas várias combinações de passos espectrais diferentes, assim como de buscas lineares não monótonas diferentes. Dos resultados numéricos apresentados em vários trabalhos não é possível inferir se existem diferenças significativas no desempenho dos diversos métodos. Além disso, também não fica clara a relevância das buscas não monótonas como uma ferramenta em si próprias ou se, na verdade, elas são úteis apenas para permitir que o método seja o mais parecido possível com o método original de Barzilai e Borwein. O objetivo deste trabalho é comparar os diversos métodos recentemente introduzidos como combinações de diferentes buscas lineares não monótonas e diferentes passos espectrais para encontrar a melhor combinação e, a partir daí, aferir o desempenho numérico do método. / The Spectral Gradient method, introduced by Barzilai and Borwein and analized by Raydan for unconstrained minimization, is a simple method whose performance is comparable to traditional methods, such as conjugate gradients. Since the introduction of method, as well as its extension to minimization of convex sets, there were introduced various combinations of different spectral steplengths, as well as different nonmonotone line searches. By the numerical results presented in many studies it is not possible to infer whether there are siginificant differences in the performance of various methods. It also is not sure the relevance of the nonmonotone line searches as a tool in themselves or whether, in fact, they are usefull only to allow the method to be as similar as possible with the original method of Barzilai e Borwein. The objective of this study is to compare the different methods recently introduced as different combinations of nonmonotone linear searches and different spectral steplengths to find the best combination and from there, evaluating the numerical performance of the method.

O objetivo geral deste estudo é o de trabalhar o problema de corte bidimensional guilhotinado irrestrito e no-estagiado. Para tanto, foram revistas regras e heurísticas a serem utilizadas e foi sugerida uma combinação da regra de simetria com a heurística de geraçgo dos pontos de cortes possíveis. Uma abordagem em grafo-E/OU, com a utilizaçgo de uma estratégia híbrida, que combina as técnicas \"Hill-Climbing\" e \"Depth-First\" para a busca em grafo, foi utilizada para a resolução do problema. Finalmente, foram comparados os resultados obtidos com resultados apresentados na literatura. / The general purpose of this study is to deal with the non-staged unconstrained guilhotined cutting problem. For this, rules and heuristics in use are reviewed and a combination between simmetry rule and possible cutting points generation heuristics in suggested. An AND/OR-graph approach, with used a hibrid strategy that combines Hill-Climbing and Depth-First techniques, in order to serach the graph, was used to solve this problem. Finally, results from this study and results presented in the literature are compared.

O objetivo geral deste estudo é o de trabalhar o problema de corte bidimensional guilhotinado irrestrito e no-estagiado. Para tanto, foram revistas regras e heurísticas a serem utilizadas e foi sugerida uma combinação da regra de simetria com a heurística de geraçgo dos pontos de cortes possíveis. Uma abordagem em grafo-E/OU, com a utilizaçgo de uma estratégia híbrida, que combina as técnicas \"Hill-Climbing\" e \"Depth-First\" para a busca em grafo, foi utilizada para a resolução do problema. Finalmente, foram comparados os resultados obtidos com resultados apresentados na literatura. / The general purpose of this study is to deal with the non-staged unconstrained guilhotined cutting problem. For this, rules and heuristics in use are reviewed and a combination between simmetry rule and possible cutting points generation heuristics in suggested. An AND/OR-graph approach, with used a hibrid strategy that combines Hill-Climbing and Depth-First techniques, in order to serach the graph, was used to solve this problem. Finally, results from this study and results presented in the literature are compared.

Orientador: Sandra Augusta Santos / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T00:20:47Z (GMT). No. of bitstreams: 1 Nascimento_IvanXavierMourado_M.pdf: 5587602 bytes, checksum: 769fbf124a59d55361b184a6ec802f66 (MD5) Previous issue date: 2014 / Resumo: Métodos de região de confiança formam uma classe de algoritmos iterativos amplamente utilizada em problemas de otimização não linear irrestrita para os quais as derivadas da função objetivo não estão disponíveis ou são imprecisas. Uma das abordagens clássicas desses métodos envolve a otimização de modelos polinomiais aproximadores para a função objetivo, construídos a cada iteração com base em conjuntos amostrais de pontos. Em um trabalho recente, Scheinberg e Toint [SIAM Journal on Optimization, 20 (6) (2010), pp. 3512-3532 ] mostram que apesar do controle do posicionamento dos pontos amostrais ser essencial para a convergência do método, é possível que tal controle ocorra de modo direto apenas no estágio final do algoritmo. Baseando-se nessas ideias e incorporando-as a um esquema algorítmico teórico, os autores investigam analiticamente uma curiosa propriedade de autocorreção da geometria dos pontos, a qual se evidencia nas iterações de insucesso. A convergência global do novo algoritmo é, então, obtida como uma consequência da geometria autocorretiva. Nesta dissertação estudamos o posicionamento dos pontos em métodos baseados em modelos quadráticos de interpolação e analisamos o desempenho computacional do algoritmo teórico proposto por Scheinberg e Toint, cujos parâmetros são determinados / Abstract: Trust-region methods are a class of iterative algorithms widely applied to nonlinear unconstrained optimization problems for which derivatives of the objective function are unavailable or inaccurate. One of the classical approaches involves the optimization of a polynomial model for the objective function, built at each iteration and based on a sample set. In a recent work, Scheinberg and Toint [SIAM Journal on Optimization, 20 (6) (2010), pp. 3512¿3532 ] proved that, despite being essential for convergence results, the improvement of the geometry (poisedness) of the sample set might occur only in the final stage of the algorithm. Based on these ideas and incorporating them into a theoretical algorithm framework, the authors investigate analytically an interesting self-correcting geometry mechanism of the interpolating set, which becomes evident at unsuccessful iterations. Global convergence for the new algorithm is then proved as a consequence of this self-correcting property. In this work we study the positioning of the sample points within interpolation-based methods that rely on quadratic models and investigate the computational performance of the theoretical algorithm proposed by Scheinberg and Toint, whose parameters are based upon either choices of previous works or numerical experiments / Mestrado / Matematica Aplicada / Mestre em Matemática Aplicada

O método do Gradiente Espectral, introduzido por Barzilai e Borwein e analisado por Raydan, para minimização irrestrita, é um método simples cujo desempenho é comparável ao de métodos tradicionais como, por exemplo, gradientes conjugados. Desde a introdução do método, assim como da sua extensão para minimização em conjuntos convexos, foram introduzidas várias combinações de passos espectrais diferentes, assim como de buscas lineares não monótonas diferentes. Dos resultados numéricos apresentados em vários trabalhos não é possível inferir se existem diferenças significativas no desempenho dos diversos métodos. Além disso, também não fica clara a relevância das buscas não monótonas como uma ferramenta em si próprias ou se, na verdade, elas são úteis apenas para permitir que o método seja o mais parecido possível com o método original de Barzilai e Borwein. O objetivo deste trabalho é comparar os diversos métodos recentemente introduzidos como combinações de diferentes buscas lineares não monótonas e diferentes passos espectrais para encontrar a melhor combinação e, a partir daí, aferir o desempenho numérico do método. / The Spectral Gradient method, introduced by Barzilai and Borwein and analized by Raydan for unconstrained minimization, is a simple method whose performance is comparable to traditional methods, such as conjugate gradients. Since the introduction of method, as well as its extension to minimization of convex sets, there were introduced various combinations of different spectral steplengths, as well as different nonmonotone line searches. By the numerical results presented in many studies it is not possible to infer whether there are siginificant differences in the performance of various methods. It also is not sure the relevance of the nonmonotone line searches as a tool in themselves or whether, in fact, they are usefull only to allow the method to be as similar as possible with the original method of Barzilai e Borwein. The objective of this study is to compare the different methods recently introduced as different combinations of nonmonotone linear searches and different spectral steplengths to find the best combination and from there, evaluating the numerical performance of the method.

京都大学 / 新制・課程博士 / 博士(情報学) / 甲第25439号 / 情博第877号 / 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 梅野 健, 准教授 加嶋 健司 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

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>

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.

Le nombre important de véhicules sur le réseau routier peut entraîner des problèmes d'encombrement et de sécurité. Les usagers des réseaux routiers qui nous intéressent sont les camionneurs qui transportent des marchandises, pouvant rouler avec des véhicules non conformes ou emprunter des routes interdites pour gagner du temps. Le transport de matières dangereuses est réglementé et certains lieux, surtout les ponts et les tunnels, leur sont interdits d'accès. Pour aider à faire appliquer les lois en vigueur, il existe un système de contrôles routiers composé de structures fixes et de patrouilles mobiles. Le déploiement stratégique de ces ressources de contrôle mise sur la connaissance du comportement des camionneurs que nous allons étudier à travers l'analyse de leurs choix de routes. Un problème de choix de routes peut se modéliser en utilisant la théorie des choix discrets, elle-même fondée sur la théorie de l'utilité aléatoire. Traiter ce type de problème avec cette théorie est complexe. Les modèles que nous utiliserons sont tels, que nous serons amenés à faire face à des problèmes de corrélation, puisque plusieurs routes partagent probablement des arcs. De plus, puisque nous travaillons sur le réseau routier du Québec, le choix de routes peut se faire parmi un ensemble de routes dont le nombre est potentiellement infini si on considère celles ayant des boucles. Enfin, l'étude des choix faits par un humain n'est pas triviale. Avec l'aide du modèle de choix de routes retenu, nous pourrons calculer une expression de la probabilité qu'une route soit prise par le camionneur. Nous avons abordé cette étude du comportement en commençant par un travail de description des données collectées. Le questionnaire utilisé par les contrôleurs permet de collecter des données concernant les camionneurs, leurs véhicules et le lieu du contrôle. La description des données observées est une étape essentielle, car elle permet de présenter clairement à un analyste potentiel ce qui est accessible pour étudier les comportements des camionneurs. Les données observées lors d'un contrôle constitueront ce que nous appellerons une observation. Avec les attributs du réseau, il sera possible de modéliser le réseau routier du Québec. Une sélection de certains attributs permettra de spécifier la fonction d'utilité et par conséquent la fonction permettant de calculer les probabilités de choix de routes par un camionneur. Il devient alors possible d'étudier un comportement en se basant sur des observations. Celles provenant du terrain ne nous donnent pas suffisamment d'information actuellement et même en spécifiant bien un modèle, l'estimation des paramètres n'est pas possible. Cette dernière est basée sur la méthode du maximum de vraisemblance. Nous avons l'outil, mais il nous manque la matière première que sont les observations, pour continuer l'étude. L'idée est de poursuivre avec des observations de synthèse. Nous ferons des estimations avec des observations complètes puis, pour se rapprocher des conditions réelles, nous continuerons avec des observations partielles. Ceci constitue d'ailleurs un défi majeur. Nous proposons pour ces dernières, de nous servir des résultats des travaux de (Bierlaire et Frejinger, 2008) en les combinant avec ceux de (Fosgerau, Frejinger et Karlström, 2013). Bien qu'elles soient de nature synthétiques, les observations que nous utilisons nous mèneront à des résultats tels, que nous serons en mesure de fournir une proposition concrète qui pourrait aider à optimiser les décisions des responsables des contrôles routiers. En effet, nous avons réussi à estimer, sur le réseau réel du Québec, avec un seuil de signification de 0,05 les valeurs des paramètres d'un modèle de choix de routes discrets, même lorsque les observations sont partielles. Ces résultats donneront lieu à des recommandations sur les changements à faire dans le questionnaire permettant de collecter des données. / Using transportation roads enables workers to reach their work facilities. Security and traffic jam issues are all the more important given that the number of vehicles is always increasing and we will focus on merchandise transporters in this study. Dangerous items transportation is under strict control as it is for example forbidden for them to be carried through a tunnel or across a bridge. Some transporters may drive a vehicle that has defects or/and they may be ta\-king some forbidden roads so as to reach their destination faster. Transportation of goods is regulated by the law and there exists a control system, whose purpose is to detect frauds and to make sure controlled vehicles are in order. The strategic deployment of control resources can be based on the knowledge of transporters behaviour, which is going to be studied through their route choice analysis. The number of routes can be unbounded especially if we consider loops, which leads to a complex problem to be solved. We can also mention issues closely related to route choice problem using discrete choice models such as correlation between routes sharing links and point out the fact that human decision process is not considered something easy. A route choice problem can be modelled based on the random utility theory and as a consequence we will focus on the discrete choice models. We are going to use such model on the real road network of Quebec and we will derive an expression of the probability, for a transporter, to pick one route. We are going to explain the way we did our study. It started first by doing a data description job as we are convinced this is a step that will help other analysts to have a clear view of the data situation. Some data are network related and the corresponding attributes collected will be used to model the road network of Quebec. We will use some attributes to explain the utility function, which leads to the definition of the function that gives the probability that a user takes a given route. Once this function is fully specified, the behaviour study can be done, except that we have a set of observations that are absolutely incomplete. When observations are a gathering of data collected during a road control, the information they provide us is not enough and thus, the parameters estimation will fail. We might seem blocked but in fact, we brought the idea of using simulated observations. We are going to estimate model parameters with firstly complete observations and in order to imitate the real conditions, we then are going to use partial observations. This constitutes a main challenge and we overcome it by using the results presented in (Bierlaire et Frejinger, 2008) combined with those from (Fosgerau, Frejinger et Karlström, 2013). We will demonstrate that even though the observations used are simulated, we will deliver conclusions that can be useful for road network managers. The main results we provide in this work is that estimation can be done with a 0,05 signification level on real road network of Quebec, while the observations are incomplete. Eventually, our results should motivate network managers to improve the set of questions they use to collect data as it would help them to strengthen their knowledge about the merchandise transporters and hopefully, the decision process will lead to optimized resource deployments.

Die kontinuierliche Weiterentwicklung und Verbesserung technischer Prozesse erfolgt heute auf der Basis stochastischer und deterministischer Optimierungsstrategien in Kombination mit der numerischen Simulation dieser Abläufe. Da die FE-Simulation von Umformvorgängen in der Regel sehr zeitintensiv ist, bietet sich für die Optimierung solcher Prozesse der Einsatz deterministischer Methoden an, da hier weniger Optimierungsschritte und somit auch weniger FE-Simulationen notwendig sind. Eine wichtige Anforderung an solche Optimierungsverfahren ist globale Konvergenz zu lokalen Minima, da die optimalen Parametersätze nicht immer näherungsweise bekannt sind. Die zwei wichtigsten Strategien zum Ausdehnen des beschränkten Konvergenzradius der natürlichen Optimierungsverfahren (newtonschrittbasierte Verfahren und Gradientenverfahren) sind die Line-Search-Strategie und die Trust-Region-Strategie. Die Grundlagen der Line-Search-Strategie werden aufgearbeitet und die wichtigsten Teilalgorithmen implementiert. Danach wird dieses Verfahren auf eine effiziente Kombination der Teilalgorithmen und Verfahrensparameter hin untersucht. Im Anschluss wird die Leistung eines Optimierungsverfahrens mit Line-Search-Strategie verglichen mit der eines ebenfalls implementierten Optimierungsverfahrens mit skalierter Trust-Region-Strategie. Die Tests werden nach Einfügen der implementierten Verfahren in das Programm SPC-Opt anhand der Lösung eines Quadratmittelproblems aus der Materialparameteridentifikation sowie der Formoptimierung eines Umformwerkzeugs vorgenommen.