Derivative-free optimization and filter methods to solve nonlinear constrained problems

In 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.

Derivative-free nonlinear optimization filter simplex

The 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.

Penalty fuzzy function for derivative-free optimization

Penalty and Barrier methods are normally used to solve Nonlinear Optimization Problems constrained problems. The problems appear in areas such as engineering and are often characterised by the fact that involved functions (objective and constraints) are non-smooth and/or their derivatives are not know. This means that optimization methods based on derivatives cannot net used. A Java based API was implemented, including only derivative-free optimizationmethods, to solve both constrained and unconstrained problems, which includes Penalty and Barriers methods. In this work a new penalty function, based on Fuzzy Logic, is presented. This function imposes a progressive penalization to solutions that violate the constraints. This means that the function imposes a low penalization when the violation of the constraints is low and a heavy penalisation when the violation is high. The value of the penalization is not known in beforehand, it is the outcome of a fuzzy inference engine. Numerical results comparing the proposed function with two of the classic penalty/barrier functions are presented. Regarding the presented results one can conclude that the prosed penalty function besides being very robust also exhibits a very good performance.

Improvement of a filters method in a derivative free optimization

Constraints nonlinear optimization problems can be solved using penalty or barrier functions. This strategy, based on solving the problems without constraints obtained from the original problem, have shown to be e ective, particularly when used with direct search methods. An alternative to solve the previous problems is the lters method. The lters method introduced by Fletcher and Ley er in 2002, , has been widely used to solve problems of the type mentioned above. These methods use a strategy di erent from the barrier or penalty functions. The previous functions de ne a new one that combine the objective function and the constraints, while the lters method treat optimization problems as a bi-objective problems that minimize the objective function and a function that aggregates the constraints. Motivated by the work of Audet and Dennis in 2004, using lters method with derivative-free algorithms, the authors developed works where other direct search meth- ods were used, combining their potential with the lters method. More recently. In a new variant of these methods was presented, where it some alternative aggregation restrictions for the construction of lters were proposed. This paper presents a variant of the lters method, more robust than the previous ones, that has been implemented with a safeguard procedure where values of the function and constraints are interlinked and not treated completely independently.

Analysis of diffusion process in fractured reservoirs using fractional derivative approach

The fractal geometry is used to model of a naturally fractured reservoir and the concept of fractional derivative is applied to the diffusion equation to incorporate the history of fluid flow in naturally fractured reservoirs. The resulting fractally fractional diffusion (FFD) equation is solved analytically in the Laplace space for three outer boundary conditions. The analytical solutions are used to analyze the response of a naturally fractured reservoir considering the anomalous behavior of oil production. Several synthetic examples are provided to illustrate the methodology proposed in this work and to explain the diffusion process in fractally fractured systems.

What is a fractional derivative?

This paper discusses the concepts underlying the formulation of operators capable of being interpreted as fractional derivatives or fractional integrals. Two criteria for required by a fractional operator are formulated. The Grünwald–Letnikov, Riemann–Liouville and Caputo fractional derivatives and the Riesz potential are accessed in the light of the proposed criteria. A Leibniz rule is also obtained for the Riesz potential.

Strange Dynamics in a Fractional Derivative of Complex-Order Network of Chaotic Oscillators

We study the peculiar dynamical features of a fractional derivative of complex-order network. The network is composed of two unidirectional rings of cells, coupled through a "buffer" cell. The network has a Z3 × Z5 cyclic symmetry group. The complex derivative Dα±jβ, with α, β ∈ R+ is a generalization of the concept of integer order derivative, where α = 1, β = 0. Each cell is modeled by the Chen oscillator. Numerical simulations of the coupled cell system associated with the network expose patterns such as equilibria, periodic orbits, relaxation oscillations, quasiperiodic motion, and chaos, in one or in two rings of cells. In addition, fixing β = 0.8, we perceive differences in the qualitative behavior of the system, as the parameter c ∈ [13, 24] of the Chen oscillator and/or the real part of the fractional derivative, α ∈ {0.5, 0.6, 0.7, 0.8, 0.9, 1.0}, are varied. Some patterns produced by the coupled system are constrained by the network architecture, but other features are only understood in the light of the internal dynamics of each cell, in this case, the Chen oscillator. What is more important, architecture and/or internal dynamics?

Aplicação de algoritmos genéticos para sintonia de controladores

Mestrado em Engenharia Electrotécnica e de Computadores

Produção de bioetanol a partir de um resíduo orgânico proveniente da central de compostagem da LIPOR

Mestrado em Engenharia Química

Controlo de águas do processo e optimização da produção de papel

Na Fábrica de Papel da Ponte Redonda fabricam-se sacos de papel multi-folhas e papel reciclado do tipo Kraft. Tendo em consideração a primeira actividade, é de grande interesse optimizar o processo de fabrico de papel com vista a incorporara a máxima taxa de papel produzido internamente nas diferentes camadas dos sacos de papel. Os papéis de maior interesse são os do tipo Fluting e Liners, tendo sido produzidos em 2010 um total de 4,9 mil toneladas, ou seja 90% de todo o papel fabricado em 2010, correspondendo a a 4 mil toneladas de papéis do tipo Liners e 0,9 mil toneladas para os papéis do tipo Fluting. Nos papéis do tipo Liners incluem-se os papéis do tipo Test-Liner e Kraft-Liner, representando em termos produtivos valores idênticos. No âmbito deste trabalho, em que se pretendeu controlar as águas do processo e optimizar a produção de papel, foram introduzidos uma unidade de flutuação e um sistema que permitisse regular a consistência da suspensão fibrosa à entrada da máquina do papel, e foram ainda estudadas as possibilidades de adição de produtos químicos para melhorar as características da pasta assim como um tratamento microbiológico mais eficaz para todo o processo. Para se avaliar se as medidas implementadas teriam um impacto positivo na qualidade desses dois tipos de papéis, desenvolveu-se o trabalho em duas fases: a primeira envolve a introdução de um sistema de flutuação e de um sistema de controlo de consistência da pasta, assim como a selecção de produtos químicos a adicionar ao processo. A segunda fase consistiu na avaliação do efeito destas medidas nas características do papel fabricado. Para o efeito foram escolhidos dois tipos de papel e de diferentes gramagens, nomeadamente Test-Liner de 80 g/m2 e Fluting de 110 g/m2. Introduziu-se um flutuador com o objectivo de tratar parte das águas do processo de fabrico com vista a serem reutilizadas em determinadas aplicações possíveis para a qualidade da água obtida (lavagens e água do processo), de modo a conseguir-se uma poupança de água, assim como aproveitar-se as lamas resultantes, ricas em fibra de celulose, para utilizá-las como matéria-prima. Foi introduzido um regulador de consistência no processo da Ponte Redonda com o objectivo de alimentar de uma forma constante a consistência da pasta à entrada da máquina do papel proporcionando uma melhor formação da folha, devido à ligação entre fibras, na direcção máquina e direcção transversal. Esse sistema inovador é um Regulador de Consistência que vem proporcionar à máquina do papel uma alimentação em fibra mais constante. O fabrico de papel apenas a partir de fibras de celulose não permitirá obter um papel com as características desejadas para a sua utilização. Para corrigir estas deficiências, são adicionados produtos químicos para atribuir ou melhorar as propriedades dos papéis. Desta forma considerou-se interessante introduzir no processo um agente de retenção numa fase posterior à preparação da pasta e antes da chegada à máquina de papel, de forma a melhorar as características da suspensão fibrosa. Assim foi implementado um sistema cuja eficácia foi avaliada. Concluiu-se que com a sua implementação a máquina de papel apresentou melhores resultados na drenagem e na turbidez da água removida, significando uma água com menor teor de matéria suspensa e dissolvida, devido à melhor agregação das fibras dispersas na suspensão fibrosa, levando a um aumento da drenagem e consequentemente melhor eficiência das prensas e secaria. Foi também elaborado um estudo para introdução de um sistema de tratamento microbiológico em todo o processo de fabrico de papel, devido à existência de microorganismos prejudiciais ao seu fabrico. Concluiu-se que a água clarificada proveniente do flutuador apresentou qualidade aceitável para os objectivos pretendidos. No entanto, considerando a eficiência de 26,5% na remoção de sólidos suspensos será necessário mais algum tempo de utilização da água clarificada, cerca de um ano, para avaliar se esta terá algum efeito prejudicial nos equipamentos. Verificou-se que devido à existência de microrganismos em todo o processo de fabrico de papel será necessário efectuar lavagens aos tinões, tanques e circuitos com alguma regularidade, aproveitando-se as paragens do processo assim como implementar um sistema de tratamento microbiológico mais eficaz. Em resultado das medidas implementadas concluiu-se que os papéis produzidos apresentaram melhorias, tendo-se obtido melhores resultados em todos os ensaios de resistência. No papel do tipo Test-Liner destacam-se os bons resultados nos ensaios de superfície, Cobb60 e rebentamento. No caso do parâmetro do Cobb60, foi um resultado surpreendente visto que por norma este tipo de papéis reciclados não suportam este ensaio. Concluiu-se também que as medidas implementadas proporcionaram uma melhor agregação e ligação entre fibras, e melhor formação da folha na máquina do papel proporcionando aos papéis propriedades físico-mecânicas mais interessantes.

Sintonia de controladores baseada em Sistemas de Inferência Difusa

Este trabalho de pesquisa e desenvolvimento tem como fundamento principal o Conceito de Controlo por Lógica Difusa. Utilizando as ferramentas do software Matlab, foi possível desenvolver um controlador com base na inferência difusa que permitisse controlar qualquer tipo de sistema físico real, independentemente das suas características. O Controlo Lógico Difuso, do inglês “Fuzzy Control”, é um tipo de controlo muito particular, pois permite o uso simultâneo de dados numéricos com variáveis linguísticas que tem por base o conhecimento heurístico dos sistemas a controlar. Desta forma, consegue-se quantificar, por exemplo, se um copo está “meio cheio” ou “meio vazio”, se uma pessoa é “alta” ou “baixa”, se está “frio” ou “muito frio”. O controlo PID é, sem dúvida alguma, o controlador mais amplamente utilizado no controlo de sistemas. Devido à sua simplicidade de construção, aos reduzidos custos de aplicação e manutenção e aos resultados que se obtêm, este controlador torna-se a primeira opção quando se pretende implementar uma malha de controlo num determinado sistema. Caracterizado por três parâmetros de ajuste, a saber componente proporcional, integral e derivativa, as três em conjunto permitem uma sintonia eficaz de qualquer tipo de sistema. De forma a automatizar o processo de sintonia de controladores e, aproveitando o que melhor oferece o Controlo Difuso e o Controlo PID, agrupou-se os dois controladores, onde em conjunto, como poderemos constatar mais adiante, foram obtidos resultados que vão de encontro com os objectivos traçados. Com o auxílio do simulink do Matlab, foi desenvolvido o diagrama de blocos do sistema de controlo, onde o controlador difuso tem a tarefa de supervisionar a resposta do controlador PID, corrigindo-a ao longo do tempo de simulação. O controlador desenvolvido é denominado por Controlador FuzzyPID. Durante o desenvolvimento prático do trabalho, foi simulada a resposta de diversos sistemas à entrada em degrau unitário. Os sistemas estudados são na sua maioria sistemas físicos reais, que representam sistemas mecânicos, térmicos, pneumáticos, eléctricos, etc., e que podem ser facilmente descritos por funções de transferência de primeira, segunda e de ordem superior, com e sem atraso.

Copper, nickel and zinc removal by peanut hulls: batch and column studies in mono, tri-component systems and with real effluent

The main goal of this research study was the removal of Cu(II), Ni(II) and Zn(II) from aqueous solutions using peanut hulls. This work was mainly focused on the following aspects: chemical characterization of the biosorbent, kinetic studies, study of the pH influence in mono-component systems, equilibrium isotherms and column studies, both in mono and tri-component systems, and with a real industrial effluent from the electroplating industry. The chemical characterization of peanut hulls showed a high cellulose (44.8%) and lignin (36.1%) content, which favours biosorption of metal cations. The kinetic studies performed indicate that most of the sorption occurs in the first 30 min for all systems. In general, a pseudo-second order kinetics was followed, both in mono and tri-component systems. The equilibrium isotherms were better described by Freundlich model in all systems. Peanut hulls showed higher affinity for copper than for nickel and zinc when they are both present. The pH value between 5 and 6 was the most favourable for all systems. The sorbent capacity in column was 0.028 and 0.025 mmol g-1 for copper, respectively in mono and tri-component systems. A decrease of capacity for copper (50%) was observed when dealing with the real effluent. The Yoon-Nelson, Thomas and Yan’s models were fitted to the experimental data, being the latter the best fit.

Determination of tartaric acid in wines by FIA with tubular tartrate-selective electrodes

A flow injection analysis (FIA) system comprising a tartrate- (TAT) selective electrode has been developed for determination of tartaric acid in wines. Several electrodes constructed for this purpose had a PVC membrane with a complex of quaternary ammonium and TAT as anion exchanger, a phenol derivative as additive, and a more or less polar mediator solvent. Characterization of the electrodes showed behavior was best for membranes with o-nitrophenyl octyl ether as solvent. On injection of 500 μL into a phosphate buffer carrier (pH = 3.1; ionic strength 10–2 mol/L) flowing at 3 mL/min, the slope was 58.06 ± 0.6 with a lower limit of linear range of 5.0 × 10–4 mol/L TAT and R2 = 0.9989. The interference of several species, e.g. chloride, bromide, iodide, nitrate, gallic acid, tannin, sucrose, glucose, fructose, acetate, and citrate, was evaluated in terms of potentiometric selectivity coefficients. The Hofmeister series was followed for inorganic species and the most interfering organic ion was citrate. When red and white wines were analyzed and the results compared with those from an independent method they were found to be accurate, with relative standard deviations below 5.0%.

Electrocatalytic evaluation of DNA damage by superoxide radical for antioxidant capacity assessment

The integrity of DNA purine bases was herein used to evaluate the antioxidant capacity. Unlike other DNA-based antioxidant sensors reported so far, the damaging agent chosen was the O 2 radical enzymatically generated by the xanthine/xanthine oxidase system. An adenine-rich oligonucleotide was adsorbed on carbon paste electrodes and subjected to radical damage in the presence/absence of several antioxidant compounds. As a result, partial damage on DNA was observed. A minor product of the radical oxidation was identified by cyclic voltammetry as a diimine adenine derivative also formed during the electrochemical oxidation of adenine/guanine bases. The protective efficiency of several antioxidant compounds was evaluated after electrochemical oxidation of the remaining unoxidized adenine bases, by measuring the electrocatalytic current of NADH mediated by the adsorbed catalyst species generated. A comparison between O 2 and OH radicals as a source of DNA lesions and the scavenging efficiency of various antioxidant compounds against both of them is discussed. Finally, the antioxidant capacity of beverages was evaluated and compared with the results obtained with an optical method.

Fractional model for malaria transmission under control strategies

We study a fractional model for malaria transmission under control strategies.Weconsider the integer order model proposed by Chiyaka et al. (2008) in [15] and modify it to become a fractional order model. We study numerically the model for variation of the values of the fractional derivative and of the parameter that models personal protection, b. From observation of the figures we conclude that as b is increased from 0 to 1 there is a corresponding decrease in the number of infectious humans and infectious mosquitoes, for all values of α. This means that this result is invariant for variation of fractional derivative, in the values tested. These results are in agreement with those obtained in Chiyaka et al.(2008) [15] for α = 1.0 and suggest that our fractional model is epidemiologically wellposed.