Browsing by Author "Correia, Aldina"
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- Adaptive Penalty and Barrier function based on Fuzzy LogicPublication . Matias, João; Correia, Aldina; Mestre, Pedro; Serodio, Carlos; Couto, Pedro; Teixeira, Christophe; Melo-Pinto, PedroOptimization methods have been used in many areas of knowledge, such as Engineering, Statistics, Chemistry, among others, to solve optimization problems. In many cases it is not possible to use derivative methods, due to the characteristics of the problem to be solved and/or its constraints, for example if the involved functions are non-smooth and/or their derivatives are not know. To solve this type of problems a Java based API has been implemented, which includes only derivative-free optimization methods, and that can be used to solve both constrained and unconstrained problems. For solving constrained problems, the classic Penalty and Barrier functions were included in the API. In this paper a new approach to Penalty and Barrier functions, based on Fuzzy Logic, is proposed. Two penalty functions, that impose a progressive penalization to solutions that violate the constraints, are discussed. The implemented functions impose a low penalization when the violation of the constraints is low and a heavy penalty when the violation is high. Numerical results, obtained using twenty-eight test problems, comparing the proposed Fuzzy Logic based functions to six of the classic Penalty and Barrier functions are presented. Considering the achieved results, it can be concluded that the proposed penalty functions besides being very robust also have a very good performance.
- 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.
- 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.
- Consumer Default Risk Assessment in a Banking InstitutionPublication . Silva, Eliana Costa e; Lopes, Isabel Cristina; Correia, Aldina; Faria, SusanaCredit scoring is an application of financial risk forecasting to consumer lending. In this study, statistical analysis is applied to credit scoring data from a financial institution to evaluate the default risk of consumer loans. The default risk was found to be influenced by the spread, the age of the consumer, the number of credit cards owned by the consumer. A lower spread, a higher number of credit cards and a younger age of the borrower are factors that decrease the risk of default. Clients receiving the salary in the same banking institution of the loan have less chances of default than clients receiving their salary in another institution. We also found that clients in the lowest income tax echelon have more propensity to default.
- 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.
- 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.
- Diana Machado, Aldina Correia and Vítor Braga Women's Entrepreneurship and InternationalizationPublication . Machado, Diana; Correia, Aldina; Braga, Vitor
- 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.
- 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.
- Female Entrepreneurship: A Grouped Division of Europe and Central AsiaPublication . Machado, Diana; Correia, Aldina; Braga, VitorThe constant changes in sociocultural conditions in the global market and the tremendous growth in the number of women's companies culminates in the growing interest in research on female entrepreneurship (Brush, 1992; Moreira et al., 2019). The main objective of this study is to contribute to a better understanding of how the economies of Europe and Central Asia group in terms of female entrepreneurship. The research was based on a quantitative analysis of data from companies belonging to 35 countries in Europe and Central Asia, using a multivariate technique, namely Cluster Analysis. This study has as an implication a better understanding of how the economies of Europe and Central Asia are distributed in terms of female entrepreneurship. The results of the study suggest that the economies of Europe and Central Asia can be divided into two groups and the difference between the two clusters can be justified by the role of women in the countries concerned.