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.

ECG denoising based on adaptive signal processing technique

An Electrocardiogram (ECG) monitoring system deals with several challenges related with noise sources. The main goal of this text was the study of Adaptive Signal Processing Algorithms for ECG noise reduction when applied to real signals. This document presents an adaptive ltering technique based on Least Mean Square (LMS) algorithm to remove the artefacts caused by electromyography (EMG) and power line noise into ECG signal. For this experiments it was used real noise signals, mainly to observe the di erence between real noise and simulated noise sources. It was obtained very good results due to the ability of noise removing that can be reached with this technique. A recolha de sinais electrocardiogr a cos (ECG) sofre de diversos problemas relacionados com ru dos. O objectivo deste trabalho foi o estudo de algoritmos adaptativos para processamento digital de sinal, para redu c~ao de ru do em sinais ECG reais. Este texto apresenta uma t ecnica de redu c~ao de ru do baseada no algoritmo Least Mean Square (LMS) para remo c~ao de ru dos causados quer pela actividade muscular (EMG) quer por ru dos causados pela rede de energia el ectrica. Para as experiencias foram utilizados ru dos reais, principalmente para aferir a diferen ca de performance do algoritmo entre os sinais reais e os simulados. Foram conseguidos bons resultados, essencialmente devido as excelentes caracter sticas que esta t ecnica tem para remover ru dos.

Square-wave voltametric method for determination of molinate concentration in a biological process using a hanging mercury drop electrode

A square-wave voltammetric (SWV) method using a hanging mercury drop electrode (HMDE) has been developed for determination of the herbicide molinate in a biodegradation process. The method is based on controlled adsorptive accumulation of molinate for 10 s at a potential of -0.8 V versus AgCl/Ag. An anodic peak, due to oxidation of the adsorbed pesticide, was observed in the cyclic voltammogram at ca. -0.320 V versus AgCl/Ag; a very small cathodic peak was also detected. The SWV calibration plot was established to be linear in the range 5.0x10-6 to 9.0x10-6 mol L-1; this corresponded to a detection limit of 3.5x10-8 mol L-1. This electroanalytical method was used to monitor the decrease of molinate concentration in river waters along a biodegradation process using a bacterial mixed culture. The results achieved with this voltammetric method were compared with those obtained by use of a chromatographic method (HPLC–UV) and no significant statistical differences were observed.

Square-wave adsorptive-stripping voltammetric detection in the quality control of fluoxetine

Electroanalytical methods based on square-wave adsorptive-stripping voltammetry (SWAdSV) and flow-injection analysis with square-wave adsorptive-stripping voltammetric detection (FIA-SWAdSV) were developed for the determination of fluoxetine (FXT). The methods were based on the reduction of FXT at a mercury drop electrode at -1.2 V versus Ag/AgCl, in a phosphate buffer of pH 12.0, and on the possibility of accumulating the compound at the electrode surface. The SWAdSV method was successfully applied in the quantification of FXT in pharmaceutical products, human serum samples, and in drug dissolution studies. Because the presence of dissolved oxygen did not interfere significantly with the analysis, it was possible to quantify FXT in several pharmaceutical products using FIA-SWAdSV. This method enables analysis of up to 120 samples per hour at reduced costs.

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.

Detection of quasi-periodic processes in complex systems: how do we quantitatively describe their properties?

It has been shown that in reality at least two general scenarios of data structuring are possible: (a) a self-similar (SS) scenario when the measured data form an SS structure and (b) a quasi-periodic (QP) scenario when the repeated (strongly correlated) data form random sequences that are almost periodic with respect to each other. In the second case it becomes possible to describe their behavior and express a part of their randomness quantitatively in terms of the deterministic amplitude–frequency response belonging to the generalized Prony spectrum. This possibility allows us to re-examine the conventional concept of measurements and opens a new way for the description of a wide set of different data. In particular, it concerns different complex systems when the ‘best-fit’ model pretending to be the description of the data measured is absent but the barest necessity of description of these data in terms of the reduced number of quantitative parameters exists. The possibilities of the proposed approach and detection algorithm of the QP processes were demonstrated on actual data: spectroscopic data recorded for pure water and acoustic data for a test hole. The suggested methodology allows revising the accepted classification of different incommensurable and self-affine spatial structures and finding accurate interpretation of the generalized Prony spectroscopy that includes the Fourier spectroscopy as a partial case.

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.

Time domain design of fractional differintegrators using least-squares

In this paper we propose the use of the least-squares based methods for obtaining digital rational approximations (IIR filters) to fractional-order integrators and differentiators of type sα, α∈R. Adoption of the Padé, Prony and Shanks techniques is suggested. These techniques are usually applied in the signal modeling of deterministic signals. These methods yield suboptimal solutions to the problem which only requires finding the solution of a set of linear equations. The results reveal that the least-squares approach gives similar or superior approximations in comparison with other widely used methods. Their effectiveness is illustrated, both in the time and frequency domains, as well in the fractional differintegration of some standard time domain functions.