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Journal ArticleDOI

Penalty Function Methods for Constrained Optimization with Genetic Algorithms

01 Apr 2005-Mathematical & Computational Applications (Association for Scientific Research)-Vol. 10, Iss: 1, pp 45-56

TL;DR: These penalty-based methods for handling constraints in Genetic Algorithms are presented and discussed and their strengths and weaknesses are discussed.

AbstractGenetic Algorithms are most directly suited to unconstrained optimization. Application of Genetic Algorithms to constrained optimization problems is often a challenging effort. Several methods have been proposed for handling constraints. The most common method in Genetic Algorithms to handle constraints is to use penalty functions. In this paper, we present these penalty-based methods and discuss their strengths and weaknesses.

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Citations
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Book
17 Feb 2014
TL;DR: This book can serve as an introductory book for graduates, doctoral students and lecturers in computer science, engineering and natural sciences, and researchers and engineers as well as experienced experts will also find it a handy reference.
Abstract: Nature-Inspired Optimization Algorithms provides a systematic introduction to all major nature-inspired algorithms for optimization. The book's unified approach, balancing algorithm introduction, theoretical background and practical implementation, complements extensive literature with well-chosen case studies to illustrate how these algorithms work. Topics include particle swarm optimization, ant and bee algorithms, simulated annealing, cuckoo search, firefly algorithm, bat algorithm, flower algorithm, harmony search, algorithm analysis, constraint handling, hybrid methods, parameter tuning and control, as well as multi-objective optimization. This book can serve as an introductory book for graduates, doctoral students and lecturers in computer science, engineering and natural sciences. It can also serve a source of inspiration for new applications. Researchers and engineers as well as experienced experts will also find it a handy reference.Discusses and summarizes the latest developments in nature-inspired algorithms with comprehensive, timely literatureProvides a theoretical understanding as well as practical implementation hintsProvides a step-by-step introduction to each algorithm

780 citations


Journal ArticleDOI
TL;DR: The ABC algorithm is applied to engineering design problems by extending the basic ABC algorithm simply by adding a constraint handling technique into the selection step of the ABC algorithm in order to prefer the feasible regions of entire search space.
Abstract: Engineering design problems are generally large scale or nonlinear or constrained optimization problems. The Artificial Bee Colony (ABC) algorithm is a successful tool for optimizing unconstrained problems. In this work, the ABC algorithm is used to solve large scale optimization problems, and it is applied to engineering design problems by extending the basic ABC algorithm simply by adding a constraint handling technique into the selection step of the ABC algorithm in order to prefer the feasible regions of entire search space. Nine well-known large scale unconstrained test problems and five well-known constrained engineering problems are solved by using the ABC algorithm and the performance of ABC algorithm is compared against those of state-of-the-art algorithms.

388 citations


Cites background from "Penalty Function Methods for Constr..."

  • ...Although the use of penalty functions is very common since its simplicity and directapplicability(SmithandCoit1997;CoelloCoello1999; Yeniay 2005; Parsopoulos and Vrahatis 2002), they have several drawbacks, too....

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Journal ArticleDOI
TL;DR: The numerical results demonstrate that constrained blended BBO outperforms SGA and performs similarly to SPSO 07 for constrained single-objective optimization problems.
Abstract: Biogeography-based optimization (BBO) is a new evolutionary optimization method that is based on the science of biogeography. We propose two extensions to BBO. First, we propose a blended migration operator. Benchmark results show that blended BBO outperforms standard BBO. Second, we employ blended BBO to solve constrained optimization problems. Constraints are handled by modifying the BBO immigration and emigration procedures. The approach that we use does not require any additional tuning parameters beyond those that are required for unconstrained problems. The constrained blended BBO algorithm is compared with solutions based on a stud genetic algorithm (SGA) and standard particle swarm optimization 2007 (SPSO 07). The numerical results demonstrate that constrained blended BBO outperforms SGA and performs similarly to SPSO 07 for constrained single-objective optimization problems.

235 citations


Cites background from "Penalty Function Methods for Constr..."

  • ...However, penalty functions have several limitations and problems which are difficult to deal with (Smith and Coit, 1997; Yeniay, 2005), including the difficulty of tuning the penalty parameters....

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Book ChapterDOI
27 Aug 2005
TL;DR: A penalty function approach is employed and the algorithm is modified to preserve feasibility of the encountered solutions to investigate the performance of the recently proposed Unified Particle Swarm Optimization method on constrained engineering optimization problems.
Abstract: We investigate the performance of the recently proposed Unified Particle Swarm Optimization method on constrained engineering optimization problems. For this purpose, a penalty function approach is employed and the algorithm is modified to preserve feasibility of the encountered solutions. The algorithm is illustrated on four well–known engineering problems with promising results. Comparisons with the standard local and global variant of Particle Swarm Optimization are reported and discussed.

210 citations


Journal ArticleDOI
TL;DR: Two novel extensions for the well known ant colony optimization (ACO) framework are introduced here, which allow the solution of mixed integer nonlinear programs (MINLPs) and a hybrid implementation based on this extended ACO framework, specially developed for complex non-convex MINLPs is presented.
Abstract: Two novel extensions for the well known ant colony optimization (ACO) framework are introduced here, which allow the solution of mixed integer nonlinear programs (MINLPs). Furthermore, a hybrid implementation (ACOmi) based on this extended ACO framework, specially developed for complex non-convex MINLPs, is presented together with numerical results. These extensions on the ACO framework have been developed to serve the needs of practitioners who face highly non-convex and computationally costly MINLPs. The performance of this new method is evaluated considering several non-convex MINLP benchmark problems and one real-world application. The results obtained by our implementation substantiate the success of this new approach.

169 citations


Cites background or methods from "Penalty Function Methods for Constr..."

  • ...A wide range of modifications of this method is known and comprehensive reviews can be found in Coello [11] or Yeniay [35]....

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  • ...adaptive or annealing, see Yeniay [35]) are more powerful and adjustable to a specific problem due to a larger number of parameters....

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  • ...death or static, see Yeniay [35]) do not require a lot of problem specific parameters to be selected, which makes their use and implementation very easy and popular....

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  • ...Sophisticated penalty methods (e.g. adaptive or annealing, see Yeniay [35]) are more powerful and adjustable to a specific problem due to a larger number of parameters....

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  • ...In general it is to note, that simple penalty methods (e.g. death or static, see Yeniay [35]) do not require a lot of problem specific parameters to be selected, which makes their use and implementation very easy and popular....

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References
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Book
01 Sep 1988
Abstract: From the Publisher: This book brings together - in an informal and tutorial fashion - the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields Major concepts are illustrated with running examples, and major algorithms are illustrated by Pascal computer programs No prior knowledge of GAs or genetics is assumed, and only a minimum of computer programming and mathematics background is required

52,793 citations


Book
01 Jan 1992
TL;DR: GAs and Evolution Programs for Various Discrete Problems, a Hierarchy of Evolution Programs and Heuristics, and Conclusions.
Abstract: 1 GAs: What Are They?.- 2 GAs: How Do They Work?.- 3 GAs: Why Do They Work?.- 4 GAs: Selected Topics.- 5 Binary or Float?.- 6 Fine Local Tuning.- 7 Handling Constraints.- 8 Evolution Strategies and Other Methods.- 9 The Transportation Problem.- 10 The Traveling Salesman Problem.- 11 Evolution Programs for Various Discrete Problems.- 12 Machine Learning.- 13 Evolutionary Programming and Genetic Programming.- 14 A Hierarchy of Evolution Programs.- 15 Evolution Programs and Heuristics.- 16 Conclusions.- Appendix A.- Appendix B.- Appendix C.- Appendix D.- References.

12,058 citations


Book
03 Mar 1993
TL;DR: The book is a solid reference for professionals as well as a useful text for students in the fields of operations research, management science, industrial engineering, applied mathematics, and also in engineering disciplines that deal with analytical optimization techniques.
Abstract: COMPREHENSIVE COVERAGE OF NONLINEAR PROGRAMMING THEORY AND ALGORITHMS, THOROUGHLY REVISED AND EXPANDED"Nonlinear Programming: Theory and Algorithms"--now in an extensively updated Third Edition--addresses the problem of optimizing an objective function in the presence of equality and inequality constraints. Many realistic problems cannot be adequately represented as a linear program owing to the nature of the nonlinearity of the objective function and/or the nonlinearity of any constraints. The "Third Edition" begins with a general introduction to nonlinear programming with illustrative examples and guidelines for model construction.Concentration on the three major parts of nonlinear programming is provided: Convex analysis with discussion of topological properties of convex sets, separation and support of convex sets, polyhedral sets, extreme points and extreme directions of polyhedral sets, and linear programmingOptimality conditions and duality with coverage of the nature, interpretation, and value of the classical Fritz John (FJ) and the Karush-Kuhn-Tucker (KKT) optimality conditions; the interrelationships between various proposed constraint qualifications; and Lagrangian duality and saddle point optimality conditionsAlgorithms and their convergence, with a presentation of algorithms for solving both unconstrained and constrained nonlinear programming problemsImportant features of the "Third Edition" include: New topics such as second interior point methods, nonconvex optimization, nondifferentiable optimization, and moreUpdated discussion and new applications in each chapterDetailed numerical examples and graphical illustrationsEssential coverage of modeling and formulating nonlinear programsSimple numerical problemsAdvanced theoretical exercisesThe book is a solid reference for professionals as well as a useful text for students in the fields of operations research, management science, industrial engineering, applied mathematics, and also in engineering disciplines that deal with analytical optimization techniques. The logical and self-contained format uniquely covers nonlinear programming techniques with a great depth of information and an abundance of valuable examples and illustrations that showcase the most current advances in nonlinear problems.

6,258 citations


Journal ArticleDOI
TL;DR: GA's population-based approach and ability to make pair-wise comparison in tournament selection operator are exploited to devise a penalty function approach that does not require any penalty parameter to guide the search towards the constrained optimum.
Abstract: Many real-world search and optimization problems involve inequality and/or equality constraints and are thus posed as constrained optimization problems. In trying to solve constrained optimization problems using genetic algorithms (GAs) or classical optimization methods, penalty function methods have been the most popular approach, because of their simplicity and ease of implementation. However, since the penalty function approach is generic and applicable to any type of constraint (linear or nonlinear), their performance is not always satisfactory. Thus, researchers have developed sophisticated penalty functions specific to the problem at hand and the search algorithm used for optimization. However, the most difficult aspect of the penalty function approach is to find appropriate penalty parameters needed to guide the search towards the constrained optimum. In this paper, GA's population-based approach and ability to make pair-wise comparison in tournament selection operator are exploited to devise a penalty function approach that does not require any penalty parameter. Careful comparisons among feasible and infeasible solutions are made so as to provide a search direction towards the feasible region. Once sufficient feasible solutions are found, a niching method (along with a controlled mutation operator) is used to maintain diversity among feasible solutions. This allows a real-parameter GA's crossover operator to continuously find better feasible solutions, gradually leading the search near the true optimum solution. GAs with this constraint handling approach have been tested on nine problems commonly used in the literature, including an engineering design problem. In all cases, the proposed approach has been able to repeatedly find solutions closer to the true optimum solution than that reported earlier.

3,096 citations


"Penalty Function Methods for Constr..." refers background in this paper

  • ...These approaches can be grouped in four major categories [28]: Category 1: Methods based on penalty functions - Death Penalty [2] - Static Penalties [15,20] - Dynamic Penalties [16,17] - Annealing Penalties [5,24] - Adaptive Penalties [10,12,35,37] - Segregated GA [21] - Co-evolutionary Penalties [8] Category 2: Methods based on a search of feasible solutions - Repairing unfeasible individuals [27] - Superiority of feasible points [9,32] - Behavioral memory [34]...

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Book
01 Jan 1996
TL;DR: In this work, the author compares the three most prominent representatives of evolutionary algorithms: genetic algorithms, evolution strategies, and evolutionary programming within a unified framework, thereby clarifying the similarities and differences of these methods.

2,595 citations