Classification of some penalty methods

Optimization problems arise in science, engineering, economy, etc. and we need to find the best solutions for each reality. The methods used to solve these problems depend on several factors, including the amount and type of accessible information, the available algorithms for solving them, and, obviously, the intrinsic characteristics of the problem. There are many kinds of optimization problems and, consequently, many kinds of methods to solve them. When the involved functions are nonlinear and their derivatives are not known or are very difficult to calculate, these methods are more rare. These kinds of functions are frequently called black box functions. To solve such problems without constraints (unconstrained optimization), we can use direct search methods. These methods do not require any derivatives or approximations of them. But when the problem has constraints (nonlinear programming problems) and, additionally, the constraint functions are black box functions, it is much more difficult to find the most appropriate method. Penalty methods can then be used. They transform the original problem into a sequence of other problems, derived from the initial, all without constraints. Then this sequence of problems (without constraints) can be solved using the methods available for unconstrained optimization. In this chapter, we present a classification of some of the existing penalty methods and describe some of their assumptions and limitations. These methods allow the solving of optimization problems with continuous, discrete, and mixing constraints, without requiring continuity, differentiability, or convexity. Thus, penalty methods can be used as the first step in the resolution of constrained problems, by means of methods that typically are used by unconstrained problems. We also discuss a new class of penalty methods for nonlinear optimization, which adjust the penalty parameter dynamically.

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.

Métodos de penalidade exacta para resolução de problemas de optimização não linear

In this work we present a classification of some of the existing Penalty Methods (denominated the Exact Penalty Methods) and describe some of its limitations and estimated. With these methods we can solve problems of optimization with continuous, discrete and mixing constrains, without requiring continuity, differentiability or convexity. The boarding consists of transforming the original problem, in a sequence of problems without constrains, derivate of the initial, making possible its resolution for the methods known for this type of problems. Thus, the Penalty Methods can be used as the first step for the resolution of constrained problems for methods typically used in by unconstrained problems. The work finishes discussing a new class of Penalty Methods, for nonlinear optimization, that adjust the penalty parameter dynamically.

Direct search optimization application programming interface with remote access

Search Optimization methods are needed to solve optimization problems where the objective function and/or constraints functions might be non differentiable, non convex or might not be possible to determine its analytical expressions either due to its complexity or its cost (monetary, computational, time,...). Many optimization problems in engineering and other fields have these characteristics, because functions values can result from experimental or simulation processes, can be modelled by functions with complex expressions or by noise functions and it is impossible or very difficult to calculate their derivatives. Direct Search Optimization methods only use function values and do not need any derivatives or approximations of them. In this work we present a Java API that including several methods and algorithms, that do not use derivatives, to solve constrained and unconstrained optimization problems. Traditional API access, by installing it on the developer and/or user computer, and remote API access to it, using Web Services, are also presented. Remote access to the API has the advantage of always allow the access to the latest version of the API. For users that simply want to have a tool to solve Nonlinear Optimization Problems and do not want to integrate these methods in applications, also two applications were developed. One is a standalone Java application and the other a Web-based application, both using the developed API.

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.

Filters method in direct search optimization, new measures to admissibility

Constrained nonlinear optimization problems are usually solved using penalty or barrier methods combined with unconstrained optimization methods. Another alternative used to solve constrained nonlinear optimization problems is the lters method. Filters method, introduced by Fletcher and Ley er in 2002, have been widely used in several areas of constrained nonlinear optimization. These methods treat optimization problem as bi-objective attempts to minimize the objective function and a continuous function that aggregates the constraint violation functions. Audet and Dennis have presented the rst lters method for derivative-free nonlinear programming, based on pattern search methods. Motivated by this work we have de- veloped a new direct search method, based on simplex methods, for general constrained optimization, that combines the features of the simplex method and lters method. This work presents a new variant of these methods which combines the lters method with other direct search methods and are proposed some alternatives to aggregate the constraint violation functions.

A web-based tool to evaluate the iterative processes of direct search methods

Constrained and unconstrained Nonlinear Optimization Problems often appear in many engineering areas. In some of these cases it is not possible to use derivative based optimization methods because the objective function is not known or it is too complex or the objective function is non-smooth. In these cases derivative based methods cannot be used and Direct Search Methods might be the most suitable optimization methods. An Application Programming Interface (API) including some of these methods was implemented using Java Technology. This API can be accessed either by applications running in the same computer where it is installed or, it can be remotely accessed through a LAN or the Internet, using webservices. From the engineering point of view, the information needed from the API is the solution for the provided problem. On the other hand, from the optimization methods researchers’ point of view, not only the solution for the problem is needed. Also additional information about the iterative process is useful, such as: the number of iterations; the value of the solution at each iteration; the stopping criteria, etc. In this paper are presented the features added to the API to allow users to access to the iterative process data.

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.

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.

Direct-search penalty/barrier methods

In Nonlinear Optimization Penalty and Barrier Methods are normally used to solve Constrained Problems. There are several Penalty/Barrier Methods and they are used in several areas from Engineering to Economy, through Biology, Chemistry, Physics among others. In these areas it often appears Optimization Problems in which the involved functions (objective and constraints) are non-smooth and/or their derivatives are not know. In this work some Penalty/Barrier functions are tested and compared, using in the internal process, Derivative-free, namely Direct Search, methods. This work is a part of a bigger project involving the development of an Application Programming Interface, that implements several Optimization Methods, to be used in applications that need to solve constrained and/or unconstrained Nonlinear Optimization Problems. Besides the use of it in applied mathematics research it is also to be used in engineering software packages.

Visualizing non-linear control system performance by means of multidimensional scaling

This paper addresses the use of multidimensional scaling in the evaluation of controller performance. Several nonlinear systems are analyzed based on the closed loop time response under the action of a reference step input signal. Three alternative performance indices, based on the time response, Fourier analysis, and mutual information, are tested. The numerical experiments demonstrate the feasibility of the proposed methodology and motivate its extension for other performance measures and new classes of nonlinearities.

Auxiliary mechanisms for telerobotics

Dissertação apresentada na Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa para obtenção do grau de Mestre em Engenharia Electrotécnica e de Computadores

Control of a power supply with cycling current using different controllers

Fast Field Cycling (FFC) Nuclear Magnetic Resonance (NMR) relaxometers require controlled current sources in order to get accurate flux density with respect to its magnet. The main elements of the proposed solution are a power semiconductor, a DC voltage source and the magnet. The power semiconductor is commanded in order to get a linear control of the flux density. To implement the flux density control, a Hall Effect sensor is used. Furthermore, the dynamic behavior of the current source is analyzed and compared when using a PI controller and a PD2I controller.

Fifth harmonic and sag impact on PMSG wind turbines with a balancing new strategy for capacitor voltages

This paper deals with the computing simulation of the impact on permanent magnet synchronous generator wind turbines due to fifth harmonic content and grid voltage decrease. Power converter topologies considered in the simulations are the two-level and the three-level ones. The three-level converters are limited by unbalance voltages in the DC-link capacitors. In order to lessen this limitation, a new control strategy for the selection of the output voltage vectors is proposed. Controller strategies considered in the simulation are respectively based on proportional integral and fractional-order controllers. Finally, a comparison between the results of the simulations with the two controller strategies is presented in order to show the main advantage of the proposed strategy. (C) 2014 Elsevier Ltd. All rights reserved.

Fractional order control on a wind turbine benchmark

This paper is about a hierarchical structure with an event-based supervisor in a higher level and a fractional-order proportional integral (FOPI) in a lower level applied to a wind turbine. The event-based supervisor analyzes the operation conditions to determine the state of the wind turbine. This controller operate in the full load region and the main objective is to capture maximum power generation while ensuring the performance and reliability required for a wind turbine to be integrated into an electric grid. The main contribution focus on the use of fractional-order proportional integral controller which benefits from the introduction of one more tuning parameter, the integral fractional-order, taking advantage over integer order proportional integral (PI) controller. Comparisons between fractional-order pitch control and a default proportional integral pitch controller applied to a wind turbine benchmark are given and simulation results by Matlab/Simulink are shown in order to prove the effectiveness of the proposed approach.