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.

Stiffness da tibiotársica em sujeitos com Acidente Vascular Encefálico - processo de raciocínio clínico

regula a posição do corpo no espaço, sendo um pré-requisito para o movimento. À periferia este processo de Controlo Postural pode ser identificado também através da variação do stiffness. O Acidente Vascular Encefálico apresenta-se como a patologia onde os sujeitos são referenciados como tendo alteração do stiffness, e poderão verificar-se modificações nesta variável no âmbito da reabilitação neuro-motora. Objetivo: Descrever o comportamento do stiffness da tibiotársica, nos dois membros inferiores, em indivíduos pós Acidente Vascular Encefálico, face a uma intervenção em fisioterapia baseada num processo de raciocínio clínico. Métodos: 5 sujeitos participaram no estudo, tendo sido implementado um programa de reabilitação para cada um dos sujeitos, por um período de 3 meses, com 2 momentos de avaliação (M0 e M1). O torque e a amplitude articular da tibiotársica foi monitorizada, através do dinamómetro isocinético, durante o movimento passivo de dorsiflexão, a diferentes velocidades (5º/s, 1º/s e 0,25º/s) A atividade eletromiográfica dos músculos Gastrocnémio Interno e Solear foi também recolhida. O valor de stiffness foi calculado através da relação torque/posição. Resultados: Em todos os sujeitos em estudo verificou-se que de uma forma geral o stiffness do membro contralateral à lesão apresentou uma modificação no sentido da diminuição em todas as amplitudes em M1. Nos sujeitos A e C, verificou-se que o stiffness do membro ipsilateral apresentou uma modificação no sentido da diminuição em M1 (em amplitudes intermédias). Nos sujeitos B, D e E o stiffness não apresentou modificações. O stiffness não variou com a velocidade. Conclusão: O stiffness apontou para uma diminuição, nos sujeitos em estudo no membro contralateral à lesão e no membro ipsilateral à lesão nos sujeitos A e C em amplitudes intermédias.

Trimethoprim-selective electrodes with molecularly imprinted polymers acting as ionophores and potentiometric transduction on graphite solid-contact

This work proposes a new biomimetic sensor material for trimethoprim. It is prepared by means of radical polymerization, having trimethylolpropane trimethacrylate as cross-linker, benzoyl peroxide as radicalar iniciator, chloroform as porogenic solvent, and methacrylic acid and 2-vinyl pyridine as monomers. Different percentages of sensor in a range between 1 and 6% were studied. Their behavior was compared to that obtained with ion-exchanger quaternary ammonium salt (additive tetrakis(p-chlorophenyl)borate or tetraphenylborate). The effect of an anionic additive in the sensing membrane was also tested. Trimethoprim sensors with 1% of imprinted particles from methacrylic acid monomers showed the best response in terms of slope (59.7 mV/decade) and detection limit (4.01×10−7 mol/L). These electrodes displayed also a good selectivity towards nickel, manganese aluminium, ammonium, lead, potassium, sodium, iron, chromium, sulfadiazine, alanine, cysteine, tryptophan, valine and glycine. The sensors were not affected by pH changes from 2 to 6. They were successfully applied to the analysis of water from aquaculture.

Observability of nonlinear fractional dynamical systems

We study the observability of linear and nonlinear fractional differential systems of order 0 < α < 1 by using the Mittag-Leffler matrix function and the application of Banach’s contraction mapping theorem. Several examples illustrate the concepts.

Combined mutation differential evolution to solve systems of nonlinear equations

This paper presents a differential evolution heuristic to compute a solution of a system of nonlinear equations through the global optimization of an appropriate merit function. Three different mutation strategies are combined to generate mutant points. Preliminary numerical results show the effectiveness of the presented heuristic.

Film thickness in a ball-on-disc contact lubricated with greases, bleed oils and base oils

Three different lubricating greases and their bleed and base oils were compared in terms of film thickness in a ball-on-disc test rig through optical interferometry. The theoretical values calculated according to Hamrock's equation are in close agreement with the base oil film thickness measurements, which validates the selected experimental methodology. The grease and bleed oil film thickness under fully flooded lubrication conditions presented quite similar behaviour and levels. Therefore, the grease film thickness under full film conditions might be predicted using their bleed oil properties, namely the viscosity and pressure-viscosity coefficient. The base and bleed oil lubricant parameter LP are proportional to the measured film thickness. A relationship between grease and the corresponding bleed oil film thickness was evidenced.

A Java expression evaluator for nonlinear programming

Finding the optimal value for a problem is usual in many areas of knowledge where in many cases it is needed to solve Nonlinear Optimization Problems. For some of those problems it is not possible to determine the expression for its objective function and/or its constraints, they are the result of experimental procedures, might be non-smooth, among other reasons. To solve such problems it was implemented an API contained methods to solve both constrained and unconstrained problems. This API was developed to be used either locally on the computer where the application is being executed or remotely on a server. To obtain the maximum flexibility both from the programmers’ and users’ points of view, problems can be defined as a Java class (because this API was developed in Java) or as a simple text input that is sent to the API. For this last one to be possible it was also implemented on the API an expression evaluator. One of the drawbacks of this expression evaluator is that it is slower than the Java native code. In this paper it is presented a solution that combines both options: the problem can be expressed at run-time as a string of chars that are converted to Java code, compiled and loaded dynamically. To wide the target audience of the API, this new expression evaluator is also compatible with the AMPL format.

Self-adaptive combination of global tabu search and local search for nonlinear equations

Solving systems of nonlinear equations is a very important task since the problems emerge mostly through the mathematical modelling of real problems that arise naturally in many branches of engineering and in the physical sciences. The problem can be naturally reformulated as a global optimization problem. In this paper, we show that a self-adaptive combination of a metaheuristic with a classical local search method is able to converge to some difficult problems that are not solved by Newton-type methods.

Web-based application programming interface to solve nonlinear optimization problems

Nonlinear 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.

Solving nonlinear equations by a tabu search strategy

Solving systems of nonlinear equations is a problem of particular importance since they emerge through the mathematical modeling of real problems that arise naturally in many branches of engineering and in the physical sciences. The problem can be naturally reformulated as a global optimization problem. In this paper, we show that a metaheuristic, called Directed Tabu Search (DTS) [16], is able to converge to the solutions of a set of problems for which the fsolve function of MATLAB® failed to converge. We also show the effect of the dimension of the problem in the performance of the DTS.

Fractional describing function of systems with nonlinear friction

This paper studies the describing function (DF) of systems consisting in a mass subjected to nonlinear friction. The friction force is composed in three components namely, the viscous, the Coulomb and the static forces. The system dynamics is analyzed in the DF perspective revealing a fractional-order behaviour. The reliability of the DF method is evaluated through the signal harmonic content and the limit cycle prediction.

Performance of fractional PID algorithms controlling nonlinear systems with saturation and backlash phenomena

The development of fractional-order controllers is currently one of the most promising fields of research. However, most of the work in this area addresses the case of linear systems. This paper reports on the analysis of fractional-order control of nonlinear systems. The performance of discrete fractional-order PID controllers in the presence of several nonlinearities is discussed. Some results are provided that indicate the superior robustness of such algorithms.

Reorganização do stiffness em indivíduos com Acidente Vascular Encefálico face à intervenção em fisioterapia. Repercussões no levantar e na marcha

Introdução: A organização estrutural e funcional do sistema nervoso face à organização dos diferentes tipos de input, no âmbito da intervenção em fisioterapia, pode potenciar um controlo postural para a regulação do stiffness e com repercussões na marcha e no levantar. Objetivo: Descrever o comportamento do stiffness da tibiotársica no movimento de dorsiflexão, no membro inferior ispi e contralesional, em indivíduos após Acidente Vascular Encefálico, face a uma intervenção em fisioterapia baseada num processo de raciocínio clínico. Pretendeu-se também observar as modificações ocorridas no âmbito da atividade electromiográfica dos flexores plantares, gastrocnémio medial e solear, durante a marcha e o levantar. Métodos: Foi implementado um programa de reabilitação em 4 indivíduos com sequelas de AVE por um período de 3 meses, tendo sido avaliados no momento inicial e final (M0 e M1). O torque e a amplitude articular da tibiotársica foi monitorizada, através do dinamómetro isocinético, durante o movimento passivo de dorsiflexão, e o nível de atividade eletromiográfica registado, através de electomiografia de superfície, no solear e gastrocnémio medial. Foram estudadas as fases de aceitação de carga no STS (fase II) e na marcha (sub-fase II). Resultados: Em todos os indivíduos em estudo verificou-se que o stiffness apresentou uma modificação no sentido da diminuição em todas as amplitudes em M1. O nível de atividade eletromiográfica teve comportamentos diferentes nos vários indivíduos. Conclusão: O stiffness apontou para uma diminuição nos indivíduos em estudo entre M0 e M1. Foram registadas modificações no nível de atividade eletromiográfica sem que seja possível identificar uma tendência clara entre os dois momentos para esta variável.

Nonlinear dynamics of the patient’s response to drug effect during general anesthesia

In today’s healthcare paradigm, optimal sedation during anesthesia plays an important role both in patient welfare and in the socio-economic context. For the closed-loop control of general anesthesia, two drugs have proven to have stable, rapid onset times: propofol and remifentanil. These drugs are related to their effect in the bispectral index, a measure of EEG signal. In this paper wavelet time–frequency analysis is used to extract useful information from the clinical signals, since they are time-varying and mark important changes in patient’s response to drug dose. Model based predictive control algorithms are employed to regulate the depth of sedation by manipulating these two drugs. The results of identification from real data and the simulation of the closed loop control performance suggest that the proposed approach can bring an improvement of 9% in overall robustness and may be suitable for clinical practice.