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.
Abstract: Genetic 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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01 May 2019
TL;DR: The results indicate that the proposed power allocation methods provide a significant improvement over the compared ones, both in terms of achieved system throughput and fairness among the users.
Abstract: The exponential proliferation of the Internet of Things (IoT) concept renders it an integral part of the emerging 5th Generation (5G) of mobile networks. In order to address the requirement of massive IoT-type connections, the spectrum has to be better utilized. Non-Orthogonal Multiple Access (NOMA) is a promising technique that can effectively improve the spectrum efficiency. In this paper, we consider the power allocation problem of a downlink Single Input Single Output (SISO) NOMA system. We aim to maximize the total system throughput, while maintaining a good level of fairness among the users. The system available bandwidth is divided into a number of subbands, and each subband is assigned to two users. The combination of the assigned users is based on the performance of each one, in terms of throughput. Two methods are reported for allocating power to each subband. In the first method, the maximum transmission power is equally divided to all bands, whereas, in the second, we use the Particle Swarm Optimization (PSO) algorithm to allocate the power to each band aiming to maximize the total system throughput. The power allocated to each band is further divided into the two multiplexed users. A series of extensive simulations and comparisons were employed to evaluate our proposed power allocation methods. The results indicate that our methods provide a significant improvement over the compared ones, both in terms of achieved system throughput and fairness among the users.
12 citations
Cites methods from "Penalty Function Methods for Constr..."
...In order to enforce the constraint of the optimization problem (5), we incorporate a penalty function to the fitness function [22] in line 11....
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TL;DR: In this paper, an ensemble optimization algorithm based on Follow The Leader (FTL), Multi-verse Optimizer (MVO), and Salp Swarm Algorithm (SSA) is proposed to solve constrained optimization problems.
12 citations
TL;DR: An indicator-based evolutionary multi-objective optimization algorithm (EMOA) for tackling Equality Constrained MOPs (ECMOPs) and can properly approximate a feasible solution by guiding the search through the use of an artificially constructed set that approximates the feasible Pareto front of a given problem.
Abstract: Many real-world applications involve dealing with several conflicting objectives which need to be optimized simultaneously. Moreover, these problems may require the consideration of limitations that restrict their decision variable space. Evolutionary Algorithms (EAs) are capable of tackling Multi-objective Optimization Problems (MOPs). However, these approaches struggle to accurately approximate a feasible solution when considering equality constraints as part of the problem due to the inability of EAs to find and keep solutions exactly at the constraint boundaries. Here, we present an indicator-based evolutionary multi-objective optimization algorithm (EMOA) for tackling Equality Constrained MOPs (ECMOPs). In our proposal, we adopt an artificially constructed reference set closely resembling the feasible Pareto front of an ECMOP to calculate the Inverted Generational Distance of a population, which is then used as a density estimator. An empirical study over a set of benchmark problems each of which contains at least one equality constraint was performed to test the capabilities of our proposed COnstrAined Reference SEt - EMOA (COARSE-EMOA). Our results are compared to those obtained by six other EMOAs. As will be shown, our proposed COARSE-EMOA can properly approximate a feasible solution by guiding the search through the use of an artificially constructed set that approximates the feasible Pareto front of a given problem.
12 citations
TL;DR: In this article, two meta-heuristic algorithms, including NSGA-II and non-dominated ranking genetic algorithm (NRGA), were proposed to solve the problem where their performances were compared in terms of four multi-objective metrics including maximum spread index (MSI), spacing, number of Pareto solution (NPS), and CPU run-time values.
Abstract: Purpose
When designing an optimization model for use in a mass casualty event response, it is common to encounter the heavy and considerable demand of injured patients and inadequate resources and personnel to provide patients with care. The purpose of this study is to create a model that is more practical in the real world. So the concept of “predicting the resource and personnel shortages” has been used in this research. Their model helps to predict the resource and personnel shortages during a mass casualty event. In this paper, to deal with the shortages, some temporary emergency operation centers near the hospitals have been created, and extra patients have been allocated to the operation center nearest to the hospitals with the purpose of improving the performance of the hospitals, reducing congestion in the hospitals and considering the welfare of the applicants.
Design/methodology/approach
The authors research will focus on where to locate health-care facilities and how to allocate the patients to multiple hospitals to take into view that in some cases of emergency situations, the patients may exceed the resource and personnel capacity of hospitals to provide conventional standards of care.
Findings
In view of the fact that the problem is high degree of complexity, two multi-objective meta-heuristic algorithms, including non-dominated sorting genetic algorithm (NSGA-II) and non-dominated ranking genetic algorithm (NRGA), were proposed to solve the model where their performances were compared in terms of four multi-objective metrics including maximum spread index (MSI), spacing (S), number of Pareto solution (NPS) and CPU run-time values. For comparison purpose, paired t-test was used. The results of 15 numerical examples showed that there is no significant difference based on MSI, S and NPS metrics, and NRGA significantly works better than NSGA-II in terms of CPU time, and the technique for the order of preference by similarity to ideal solution results showed that NRGA is a better procedure than NSGA-II.
Research limitations/implications
The planning horizon and time variable have not been considered in the model, for example, the length of patients’ hospitalization at hospitals.
Practical implications
Presenting an effective strategy to respond to a mass casualty event (natural and man-made) is the main goal of the authors’ research.
Social implications
This paper strategy is used in all of the health-care centers, such as hospitals, clinics and emergency centers when dealing with disasters and encountering with the heavy and considerable demands of injured patients and inadequate resources and personnel to provide patients with care.
Originality/value
This paper attempts to shed light onto the formulation and the solution of a three-objective optimization model. The first part of the objective function attempts to maximize the covered population of injured patients, the second objective minimizes the distance between hospitals and temporary emergency operation centers and the third objective minimizes the distance between the warehouses and temporary centers.
12 citations
01 Jan 2016
TL;DR: In this paper, a 3D finite element model for flexible rolling of a 250 μm thick workpiece with reduction of 20 to 50%, and rolling phase with thinner thickness indicates a better ability to decrease the surface roughness.
Abstract: Micro flexible rolling aims to fabricate submillimeter thick strips with varying thickness profile, where the surface quality of products is mainly determined by initial workpiece surface roughness and subsequent surface asperity flattening process, which is affected by process parameters during rolling. This paper shows a 3D finite element model for flexible rolling of a 250 μm thick workpiece with reduction of 20 to 50%, and rolling phase with thinner thickness indicates a better ability to decrease the surface roughness. Four types of initial workpiece surface roughness are studied in the simulation, and the influences of process parameters, such as friction coefficient, rolling speed and roll gap adjusting speed, on surface asperity flattening of workpieces with different initial surface roughness have been numerically investigated and analysed.
11 citations
References
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Book•
01 Sep 1988
TL;DR: In this article, the authors present the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields, including computer programming and mathematics.
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
52,797 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.
12,212 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.
6,259 citations
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.
3,495 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.
2,679 citations