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.

Influence of mixtures of acenaphthylene and benzo[a] anthracene on their degradation by Pleurotus ostreatus in sandy soil

Purpose Polycyclic aromatic hydrocarbons (PAHs) are a class of organic compounds commonly found as soil contaminants. Fungal degradation is considered as an environmentally friendly and cost-effective approach to remove PAHs from soil. Acenaphthylene (Ace) and Benzo[a]anthracene (BaA) are two PAHs that can coexist in soils; however, the influence of the presence of each other on their biodegradation has not been studied. The biodegradation of Ace and BaA, alone and in mixtures, by the white rot fungus Pleurotus ostreatus was studied in a sandy soil. Materials and methods Experimental microcosms containing soil spiked with different concentrations of Ace and BaAwere inoculated with P. ostreatus. Initial (t 0) and final (after 15 days of incubation) soil concentrations of Ace and BaA were determined after extraction of the PAHs. Results and discussion P. ostreatus was able to degrade 57.7% of the Ace in soil spiked at 30 mg kg−1 dry soil and 65.8% of Ace in soil spiked at 60 mg kg−1 dry soil. The degradation efficiency of BaA by P. ostreatus was 86.7 and 77.4% in soil spiked with Ace at 30 and 60 mg kg−1 dry soil, respectively. After 15 days of incubation, there were no significant differences in Ace concentration between soil spiked with Ace and soil spiked with Ace + BaA, irrespective of the initial soil concentration of both PAHs. There were also no differences in BaA concentration between soil spiked with BaA and soil spiked with BaA + Ace. Conclusions The results indicate that the fungal degradation of Ace and BaA was not influenced by the presence of each other’s PAH in sandy soil. Bioremediation of soils contaminated with Ace and BaA using P. ostreatus is a promising approach to eliminate these PAHs from the environment.

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

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.

Analysis of polycyclic aromatic hydrocarbons in atmospheric particulate samples by microwaveassisted extraction and liquid chromatography

A methodology based on microwave-assisted extraction (MAE) and LC with fluorescence detection (FLD) was investigated for the efficient determination of 15 polycyclic aromatic hydrocarbons (PAHs) regarded as priority pollutants by the US Environmental Protection Agency and dibenzo(a,l)pyrene in atmospheric particulate samples. PAHs were successfully extracted from real outdoor particulate matter (PM) samples with recoveries ranging from 81.4±8.8 to 112.0±1.1%, for all the compounds except for naphthalene (62.3±18.0%) and anthracene (67.3±5.7%), under the optimum MAE conditions (30.0 mL of ACN for 20 min at 110ºC). No clean-up steps were necessary prior to LC analysis. LOQs ranging from 0.0054 ng/m3 for benzo( a)anthracene to 0.089 ng/m3 for naphthalene were reached. The validated MAE methodology was applied to the determination of PAHs from a set of real world PM samples collected in Oporto (north of Portugal). The sum of particulate-bound PAHs in outdoor PM ranged from 2.5 and 28 ng/m3.

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.

Influence of traffic emissions on the carcinogenic polycyclic aromatic hydrocarbons in outdoor breathable particles

Because polycyclic aromatic hydrocarbons (PAHs) have been proven to be toxic, mutagenic, and/or carcinogenic, there is widespread interest in analyzing and evaluating exposure to PAHs in atmospheric environments influenced by different emission sources. Because traffic emissions are one of the biggest sources of fine particles, more information on carcinogenic PAHs associated with fine particles needs to be provided. Aiming to further understand the impact of traffic particulate matter (PM) on human health, this study evaluated the influence of traffic on PM10 (PM with aerodynamic diameter <10 µm) and PM2.5 (PM with aerodynamic diameter <2.5 µm), considering their concentrations and compositions in carcinogenic PAHs. Samples were collected at one site influenced by traffic emissions and at one reference site using lowvolume samplers. Analysis of PAHs was performed by microwave-assisted extraction combined with liquid chromatography (MAE-LC); 17 PAHs, including 9 carcinogenic ones, were quantified. At the site influenced by traffic emissions, PM10 and PM2.5 concentrations were, respectively, 380 and 390% higher than at the background site. When influenced by traffic emissions, the total concentration of nine carcinogenic compounds (naphthalene, chrysene, benzo(a)anthracene, benzo(b) fluoranthene, benzo(k)fluoranthene, benzo(a)pyrene, dibenzo(a,h)anthracene, indeno(1,2,3-cd)pyrene, and dibenzo(a,l)pyrene) was increased by 2400 and 3000% in PM10 and PM2.5, respectively; these nine carcinogenic compounds represented 68 and 74% of total PAHs (ƩPAHs) for PM10 and PM2.5, respectively. All PAHs, including the carcinogenic compounds, were mainly present in fine particles. Considering the strong influence of these fine particles on human health, these conclusions are relevant for the development of strategies to protect public health.

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.