Um método estocástico para resolver problemas MINLP não suaves

Tese de Doutoramento em Ciências (especialidade em Matemática)Muitos dos problemas de otimização que surgem com frequência numa vasta gama de aplicações reais, pertencem à área da Otimização Não Linear. Em geral, estes problemas são complexos, pois podem, eventualmente, incluir variáveis contínuas e inteiras, funções não lineares, não contínuas e não diferenciáveis. Nesta área existem duas classes de pro- blemas considerados de difícil resolução aos quais a comunidade científica se tem dedicado: os problemas de programação não linear (NLP) e os problemas de programação não linear inteira-mista (MINLP), não convexos e não suaves. Dada a sua complexidade, em muitos destes problemas não é possível calcular, nem sequer aproximar o cálculo de derivadas, sendo por isso necessário desenvolver outro tipo de abordagem diferente da convencional, para os resolver. Uma das abordagens mais promissoras e utilizadas para resolver este tipo de problemas é o uso de métodos estocásticos. Na literatura existem vários méto- dos estocásticos, baseados em populações, que têm sido usados com sucesso para resolver estes problemas. Recentemente, surgiu o Firefly Algorithm (FA) para resolver problemas de otimização contínuos e com restrições de limites simples, que se tem revelado ser bem sucedido na resolução de problemas práticos e complexos. Assim, o objetivo central desta tese prende-se com o desenvolvimento de extensões do FA capazes de resolver problemas de NLP e de MINLP, com restrições, não convexos e não suaves. Numa primeira fase, são desenvolvidos algoritmos para a resolução de problemas de NLP com restrições. São propostas duas extensões do FA baseadas em técnicas distintas, para o tratamento das restrições. Uma baseada em esquemas de ordenação dos pontos da população e outra baseada numa nova função de penalidade auto-adaptativa. Esta função de penalidade auto-adaptativa pode ser implementada em diversos métodos estocásticos de otimização global baseados em populações. O estudo da convergência do algoritmo com a função de penalidade auto-adaptativa, no contexto do FA, é demonstrada em termos probabilísticos. Numa segunda fase, foram desenvolvidos algoritmos para a resolução de problemas de MINLP. Primeiro desenvolveu-se a extensão do FA para resolver problemas de MINLP com restrições de limites simples e variáveis binárias, baseada em heurísticas. A seguir, foi desenvolvida uma extensão do FA que implementa um algoritmo de penalidade exata, para resolver os problemas de MINLP com restrições de limites simples e variáveis contínuas e inteiras. Neste contexto, foram propostas duas funções de penalidade exata, e o estudo da convergência do algoritmo de penalidade, no contexto do FA, é demonstrada em termos probabilísticos. Por fim, foi desenvolvida uma extensão do FA para resolver problemas de MINLP com restrições genéricas, que combina uma heurística e um esquema de ordenação dos pontos da população, para o tratamento das restrições do problema. As experiências numéricas realizadas mostraram que as extensões do FA propostas são eficazes, tendo-se obtido soluções de qualidade elevada, quando comparados com outros métodos disponíveis na literatura.Many optimization problems that frequently arise in a wide range of real applications belong to the area of Nonlinear Optimization. In general, these problems are complex, and may eventually involve continuous and integer variables, and nonlinear, non continuous and non differentiable functions. In this area there are two classes of optimization problems considered hard to solve, that have been studied by the scientific community: NonLinear Programming (NLP) problems and Mixed-Integer NonLinear Programming (MINLP) pro- blems, nonconvex and nonsmooth problems. Since in many of these problems it is not possible to calculate or even approximate the derivatives, it is necessary to develop a new kind of approach different from the conventional one. The stochastic methods are the most commonly used to solve these type of problems. In the literature there are several population-based stochastic methods that have been successfully used to solve these pro- blems. Recently, the Firefly Algorithm (FA) has emerged to solve continuous optimization problems with simple bounds that have proven to be successful in solving practical and complex problems. Thus, the main goal of this thesis is to extend the FA to solve constrained nonconvex and nonsmooth NLP and MINLP problems. In the first phase, algorithms to solve constrained NLP problems are developed. Two extensions of FA using different techniques for handling the constraints are proposed. One is based on ordering schemes of population points and another based on a new self-adaptive penalty function. The self-adaptive penalty function can be implemented in several population-based stochastic methods. The convergence study of the algorithm based on the self-adaptive penalty function, in the FA context, in probabilistic terms is demonstrated. In a second phase, algorithms to solve MINLP problems were developed. First, an FA extension was developed to solve MINLP problems with simple bounds and binary variables based on heuristics. Next, an FA extension that implements an exact penalty algorithm to solve the MINLP problems with simple bounds and mixed variables was developed. In this context, two exact penalty functions were proposed and the convergence study of the penalty algorithm, in the context of the FA, in probabilistic terms is proved. Finally, an FA extension to solve constrained MINLP problems, which combines a heuristic and a scheme of ordering the points of the population, for handling the constraints was developed. Numerical experiments have shown that the proposed FA extensions are effective and produced high quality solutions when compared to other methods available in the litera- ture

Extensions of firefly algorithm for nonsmooth nonconvex constrained optimization problems

Publicado em: "Computational science and its applications – ICCSA 2016: 16th International Conference, Beijing, China, July 4-7, 2016, Proceedings, Part I". ISBN 978-3-319-42084-4Firefly Algorithm (FA) is a stochastic population-based algorithm based on the flashing patterns and behavior of fireflies. Original FA was created and successfully applied to solve bound constrained optimization problems. In this paper we present extensions of FA for solving nonsmooth nonconvex constrained global optimization problems. To handle the constraints of the problem, feasibility and dominance rules and a fitness function based on the global competitive ranking, are proposed. To enhance the speed of convergence, the proposed extensions of FA invoke a stochastic local search procedure. Numerical experiments to validate the proposed approaches using a set of well know test problems are presented. The results show that the proposed extensions of FA compares favorably with other stochastic population-based methods.COMPETE: POCI-01-0145- FEDER-007043FCT – Fundação para a Ciência e Tecnologia within the projects UID/CEC/00319/2013 and UID/MAT/00013/201

A firefly dynamic penalty approach for solving engineering design problems

Firefly Algorithm is a recent swarm intelligence method, inspired by the social behavior of fireflies, based on their flashing and attraction characteristics [1, 2]. In this paper, we analyze the implementation of a dynamic penalty approach combined with the Firefly algorithm for solving constrained global optimization problems. In order to assess the applicability and performance of the proposed method, some benchmark problems from engineering design optimization are considered.The authors would like to thank the financial support from FCT (Fundação para a Ciência e Tecnologia, Portugal) in the scope of the projects: PEst-OE/MAT/UI0013/2014 and PEst-OE/EEI/UI0319/2014

Extension of the firefly algorithm and preference rules for solving MINLP problems

An extension of the firefly algorithm (FA) for solving mixed-integer nonlinear programming (MINLP) problems is presented. Although penalty functions are nowadays frequently used to handle integrality conditions and inequality and equality constraints, this paper proposes the implementation within the FA of a simple rounded-based heuristic and four preference rules to find and converge to MINLP feasible solutions. Preliminary numerical experiments are carried out to validate the proposed methodology.This work has been supported by COMPETE: POCI-01-0145-FEDER-007043 and FCT - Fundac¸ao para a Ci ˜ encia e Tecnologia, ˆ within the projects UID/CEC/00319/2013 and UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Comparison of penalty functions on a penalty approach to mixed-integer optimization

In this paper, we present a comparative study involving several penalty functions that can be used in a penalty approach for globally solving bound mixed-integer nonlinear programming (bMIMLP) problems. The penalty approach relies on a continuous reformulation of the bMINLP problem by adding a particular penalty term to the objective function. A penalty function based on the ‘erf’ function is proposed. The continuous nonlinear optimization problems are sequentially solved by the population-based firefly algorithm. Preliminary numerical experiments are carried out in order to analyze the quality of the produced solutions, when compared with other penalty functions available in the literature.The authors would like to thank the financial support from FCT (Fundação para a Ciência e Tecnologia, Portugal) in the scope of the projects: PEst-OE/MAT/UI0013/2014 and PEst-UID/CEC/00319/2013

A História da Alimentação: balizas historiográficas

Os M. pretenderam traçar um quadro da História da Alimentação, não como um novo ramo epistemológico da disciplina, mas como um campo em desenvolvimento de práticas e atividades especializadas, incluindo pesquisa, formação, publicações, associações, encontros acadêmicos, etc. Um breve relato das condições em que tal campo se assentou faz-se preceder de um panorama dos estudos de alimentação e temas correia tos, em geral, segundo cinco abardagens Ia biológica, a econômica, a social, a cultural e a filosófica!, assim como da identificação das contribuições mais relevantes da Antropologia, Arqueologia, Sociologia e Geografia. A fim de comentar a multiforme e volumosa bibliografia histórica, foi ela organizada segundo critérios morfológicos. A seguir, alguns tópicos importantes mereceram tratamento à parte: a fome, o alimento e o domínio religioso, as descobertas européias e a difusão mundial de alimentos, gosto e gastronomia. O artigo se encerra com um rápido balanço crítico da historiografia brasileira sobre o tema

Firefly penalty-based algorithm for bound constrained mixed-integer nonlinear programming

In this article, we aim to extend the firefly algorithm (FA) to solve bound constrained mixed-integer nonlinear programming (MINLP) problems. An exact penalty continuous formulation of the MINLP problem is used. The continuous penalty problem comes out by relaxing the integrality constraints and by adding a penalty term to the objective function that aims to penalize integrality constraint violation. Two penalty terms are proposed, one is based on the hyperbolic tangent function and the other on the inverse hyperbolic sine function. We prove that both penalties can be used to define the continuous penalty problem, in the sense that it is equivalent to the MINLP problem. The solutions of the penalty problem are obtained using a variant of the metaheuristic FA for global optimization. Numerical experiments are given on a set of benchmark problems aiming to analyze the quality of the obtained solutions and the convergence speed. We show that the firefly penalty-based algorithm compares favourably with the penalty algorithm when the deterministic DIRECT or the simulated annealing solvers are invoked, in terms of convergence speed.This work has been supported by FCT (Fundacao para a Ciencia e Tecnologia, Portugal) in the scope of the projects: PEst-UID/CEC/00319/2013 and PEst-OE/MAT/UI0013/2014

Impact of calcification and infrapopliteal outflow on the outcome of endovascular treatment of femoropopliteal occlusive disease

Objectives In this paper, we report the long-term outcomes of the endovascular treatment of femoropopliteal occlusive disease, focusing on the importance of calcification and runoff outflow on limb salvage and patency, and the factors associated with these outcomes at a single center. Methods This retrospective cohort study included consecutive patients with femoropopliteal occlusive who underwent femoropopliteal angioplasty at the Division of Vascular and Endovascular Surgery, Hospital do Servidor Público Estadual, São Paulo, Brazil, between January 2015 and July 2017. Results In total, 86 femoropopliteal occlusive angioplasties were performed in 86 patients, with an initial technical success rate of 95.34%. The mean ± standard deviation follow-up time was 880 ± 68.84 days. The analysis was performed at 720 days. Technical failure occurred in four patients, who were excluded from the analysis, leaving 82 patients and 82 femoropopliteal occlusive angioplasties. The estimated primary patency, secondary patency, limb salvage, and overall survival rates at 720 days were 60%, 96%, 90%, and 82.5%, respectively. In univariate and multivariate analyses, Cox regression showed worse primary patency rates in patients with one tibial vessel or isolated popliteal artery runoff ( p  = 0.005), calcification grade 4 ( p  = 0.019), calcification grade > 2 ( p  = 0.017), small vessel diameter  2, small vessel diameter < 4 mm, and no stents use. One tibial vessel or isolated popliteal artery runoff was also associated with limb loss in a univariate Cox regression analysis