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- Aplicação do Método da Transformada de Laplace a uma Classe de Operadores Integrais de ConvoluçãoPublication . Correia, AldinaRESUMO------------------------------------------------------------------------------------------------------------------------ Neste trabalho apresentamos resultados do nosso estudo sobre operadores integrais de convolução, baseado no método das transformadas de Laplace e de Mellin. Apresentamos as definições e estabelecemos condições da sua existência, em vários espaços de Lebesgue. Além disso, provamos fórmulas de inversão de certos operadores integrais de convolução do tipo de Mellin. São discutidas algumas relações entre a transformada de Laplace e a transformada Mellin. Por fim, aplicamos estes métodos na investigação de operadores de convolução de Mellin, relacionados com núcleos associados a funções de Bessel, de Gauss e de Bessel modificadas.
- Métodos de penalidade exacta para resolução de problemas de optimização não linearPublication . Correia, Aldina; Matias, João; Serôdio, CarlosIn this work we present a classification of some of the existing Penalty Methods (denominated the Exact Penalty Methods) and describe some of its limitations and estimated. With these methods we can solve problems of optimization with continuous, discrete and mixing constrains, without requiring continuity, differentiability or convexity. The boarding consists of transforming the original problem, in a sequence of problems without constrains, derivate of the initial, making possible its resolution for the methods known for this type of problems. Thus, the Penalty Methods can be used as the first step for the resolution of constrained problems for methods typically used in by unconstrained problems. The work finishes discussing a new class of Penalty Methods, for nonlinear optimization, that adjust the penalty parameter dynamically.
- Derivative-free optimization and filter methods to solve nonlinear constrained problemsPublication . Correia, Aldina; Matias, João; Mestre, Pedro; Serôdio, CarlosIn real optimization problems, usually the analytical expression of the objective function is not known, nor its derivatives, or they are complex. In these cases it becomes essential to use optimization methods where the calculation of the derivatives, or the verification of their existence, is not necessary: the Direct Search Methods or Derivative-free Methods are one solution. When the problem has constraints, penalty functions are often used. Unfortunately the choice of the penalty parameters is, frequently, very difficult, because most strategies for choosing it are heuristics strategies. As an alternative to penalty function appeared the filter methods. A filter algorithm introduces a function that aggregates the constrained violations and constructs a biobjective problem. In this problem the step is accepted if it either reduces the objective function or the constrained violation. This implies that the filter methods are less parameter dependent than a penalty function. In this work, we present a new direct search method, based on simplex methods, for general constrained optimization that combines the features of the simplex method and filter methods. This method does not compute or approximate any derivatives, penalty constants or Lagrange multipliers. The basic idea of simplex filter algorithm is to construct an initial simplex and use the simplex to drive the search. We illustrate the behavior of our algorithm through some examples. The proposed methods were implemented in Java.
- Web-based application programming interface to solve nonlinear optimization problemsPublication . Matias, João; Correia, Aldina; Mestre, Pedro; Fraga, Carlos; Serôdio, CarlosNonlinear Optimization Problems are usual in many engineering fields. Due to its characteristics the objective function of some problems might not be differentiable or its derivatives have complex expressions. There are even cases where an analytical expression of the objective function might not be possible to determine either due to its complexity or its cost (monetary, computational, time, ...). In these cases Nonlinear Optimization methods must be used. An API, including several methods and algorithms to solve constrained and unconstrained optimization problems was implemented. This API can be accessed not only as traditionally, by installing it on the developer and/or user computer, but it can also be accessed remotely using Web Services. As long as there is a network connection to the server where the API is installed, applications always access to the latest API version. Also an Web-based application, using the proposed API, was developed. This application is to be used by users that do not want to integrate methods in applications, and simply want to have a tool to solve Nonlinear Optimization Problems.
- Direct search optimization application programming interface with remote accessPublication . Mestre, Pedro; Matias, João; Correia, Aldina; Serôdio, CarlosSearch Optimization methods are needed to solve optimization problems where the objective function and/or constraints functions might be non differentiable, non convex or might not be possible to determine its analytical expressions either due to its complexity or its cost (monetary, computational, time,...). Many optimization problems in engineering and other fields have these characteristics, because functions values can result from experimental or simulation processes, can be modelled by functions with complex expressions or by noise functions and it is impossible or very difficult to calculate their derivatives. Direct Search Optimization methods only use function values and do not need any derivatives or approximations of them. In this work we present a Java API that including several methods and algorithms, that do not use derivatives, to solve constrained and unconstrained optimization problems. Traditional API access, by installing it on the developer and/or user computer, and remote API access to it, using Web Services, are also presented. Remote access to the API has the advantage of always allow the access to the latest version of the API. For users that simply want to have a tool to solve Nonlinear Optimization Problems and do not want to integrate these methods in applications, also two applications were developed. One is a standalone Java application and the other a Web-based application, both using the developed API.
- Classification of some penalty methodsPublication . Correia, Aldina; Matias, João; Mestre, Pedro; Serôdio, CarlosOptimization problems arise in science, engineering, economy, etc. and we need to find the best solutions for each reality. The methods used to solve these problems depend on several factors, including the amount and type of accessible information, the available algorithms for solving them, and, obviously, the intrinsic characteristics of the problem. There are many kinds of optimization problems and, consequently, many kinds of methods to solve them. When the involved functions are nonlinear and their derivatives are not known or are very difficult to calculate, these methods are more rare. These kinds of functions are frequently called black box functions. To solve such problems without constraints (unconstrained optimization), we can use direct search methods. These methods do not require any derivatives or approximations of them. But when the problem has constraints (nonlinear programming problems) and, additionally, the constraint functions are black box functions, it is much more difficult to find the most appropriate method. Penalty methods can then be used. They transform the original problem into a sequence of other problems, derived from the initial, all without constraints. Then this sequence of problems (without constraints) can be solved using the methods available for unconstrained optimization. In this chapter, we present a classification of some of the existing penalty methods and describe some of their assumptions and limitations. These methods allow the solving of optimization problems with continuous, discrete, and mixing constraints, without requiring continuity, differentiability, or convexity. Thus, penalty methods can be used as the first step in the resolution of constrained problems, by means of methods that typically are used by unconstrained problems. We also discuss a new class of penalty methods for nonlinear optimization, which adjust the penalty parameter dynamically.
- Direct-search penalty/barrier methodsPublication . Correia, Aldina; Matias, João; Mestre, Pedro; Serôdio, CarlosIn Nonlinear Optimization Penalty and Barrier Methods are normally used to solve Constrained Problems. There are several Penalty/Barrier Methods and they are used in several areas from Engineering to Economy, through Biology, Chemistry, Physics among others. In these areas it often appears Optimization Problems in which the involved functions (objective and constraints) are non-smooth and/or their derivatives are not know. In this work some Penalty/Barrier functions are tested and compared, using in the internal process, Derivative-free, namely Direct Search, methods. This work is a part of a bigger project involving the development of an Application Programming Interface, that implements several Optimization Methods, to be used in applications that need to solve constrained and/or unconstrained Nonlinear Optimization Problems. Besides the use of it in applied mathematics research it is also to be used in engineering software packages.
- Derivative-free nonlinear optimization filter simplexPublication . Correia, Aldina; Matias, João; Mestre, Pedro; Serôdio, CarlosThe filter method is a technique for solving nonlinear programming problems. The filter algorithm has two phases in each iteration. The first one reduces a measure of infeasibility, while in the second the objective function value is reduced. In real optimization problems, usually the objective function is not differentiable or its derivatives are unknown. In these cases it becomes essential to use optimization methods where the calculation of the derivatives or the verification of their existence is not necessary: direct search methods or derivative-free methods are examples of such techniques. In this work we present a new direct search method, based on simplex methods, for general constrained optimization that combines the features of simplex and filter methods. This method neither computes nor approximates derivatives, penalty constants or Lagrange multipliers.
- Métodos de Pesquisa Direta: Otimização não LinearPublication . Correia, AldinaOs Problemas de Optimização aparecem frequentemente em diversas áreas tais como a Engenharia, Economia, Química, entre outras. Nestas áreas aparecem usualmente Problemas onde as funções envolvidas (função objectivo e restrições) podem ser não suaves, as suas derivadas não são conhecidas, têm expressões complexas ou até casos em que as suas expressões analíticas não podem ser determinadas, seja pela sua complexidade ou pelo seu custo (monetário, computacional, temporal,...). Nestes casos os métodos que usam derivadas não são os mais apropriados para os resolver e os métodos que usam modelos para aproximar as funções mostram-se muitas vezes ineficazes. Neste trabalho estudam-se, implementam-se e comparam-se Métodos de Pesquisa Directa, isto é, métodos que usam apenas informação sobre os valores das funções, progredindo em direcção à solução óptima, comparando estes valores em determinados pontos, sem recorrer ao uso de derivadas, suas aproximações ou modelos que aproximem as funções envolvidas. Inicialmente será feita a apresentação de uma síntese sobre os métodos propostos na literatura da especialidade. Estes métodos serão posteriormente implementados e testadas algumas modificações, tendo em vista à melhoria da sua eficiência. No que respeita à Optimização sem Restrições foram estudados os métodos clássicos de Pesquisa Directa e apresentam-se novas metodologias, adoptadas de desenvolvimentos recentes nesta área, tendo os correspondentes algoritmos sido implementados, analisados e comparados. O mesmo sucedeu para os Métodos de Optimização de Problemas com Restrições, para os quais se adaptaram e apresentam alternativas de melhoria de métodos já usados na Optimização por Pesquisa Directa, como é o caso dos Métodos de Penalidade e Barreira. São também desenvolvidas técnicas que se consideram como possíveis alternativas de resolução deste tipo de problemas, como é o caso do Método dos Filtros, que dispensando a criação e uso de uma função de Penalidade/Barreira, bem como a escolha de parâmetros de penalidade, se mostrou como uma alternativa válida. A implementação destes algoritmos, com recurso à Tecnologia Java, correspondeu ao desenvolvimento de uma API que foi usada para realizar os testes numéricos e onde se encontram implementados os algoritmos e variantes aqui propostos.
- Filters method in direct search optimization, new measures to admissibilityPublication . Correia, Aldina; Matias, João; Mestre, Pedro; Serôdio, CarlosConstrained nonlinear optimization problems are usually solved using penalty or barrier methods combined with unconstrained optimization methods. Another alternative used to solve constrained nonlinear optimization problems is the lters method. Filters method, introduced by Fletcher and Ley er in 2002, have been widely used in several areas of constrained nonlinear optimization. These methods treat optimization problem as bi-objective attempts to minimize the objective function and a continuous function that aggregates the constraint violation functions. Audet and Dennis have presented the rst lters method for derivative-free nonlinear programming, based on pattern search methods. Motivated by this work we have de- veloped a new direct search method, based on simplex methods, for general constrained optimization, that combines the features of the simplex method and lters method. This work presents a new variant of these methods which combines the lters method with other direct search methods and are proposed some alternatives to aggregate the constraint violation functions.