A Population Size Reduction Approach for Nondominated Sorting-Based Optimization Algorithms

2017 ◽  
Vol 16 (01) ◽  
pp. 1750005
Author(s):  
O. Tolga Altinoz ◽  
A. Egemen Yilmaz
Keyword(s):  
Population Size ◽  
Pareto Front ◽  
Size Reduction ◽  
Benchmark Problems ◽  
Nsga Ii ◽  
General Belief ◽  
Objective Space ◽  
Overall Performance ◽  
Approximation Set ◽  
Nondominated Sorting

The solution set of any multi-objective optimization problem can be expressed as an approximation set of Pareto front. The number of solution candidates in this set could be large enough to cover the entire Pareto front as a general belief. However, among the sufficiently close points on the objective space, almost same accurate solutions can obtain. Hence, in this set, it is possible to eliminate some of the solutions without detriment to the overall performance. Therefore, in this research, the authors propose a population size reduction method which systematically reduced the population size based on the distance and angle relations between any consecutive solutions. The results are evaluated based on two-objective benchmark problems and compared with the results of NSGA-II algorithm with respect to three different performance evaluation metrics.

On Benchmark Problems and Metrics for Decision Space Performance Analysis in Multi-Objective Optimization

2017 ◽  
Vol 16 (01) ◽  
pp. 1750006 ◽  
Author(s):  
Bin Zhang ◽  
Kamran Shafi ◽  
Hussein Abbass
Keyword(s):  
Performance Analysis ◽  
Fitness Landscape ◽  
Benchmark Problems ◽  
Test Problems ◽  
Multi Objective Optimization ◽  
Decision Space ◽  
Nsga Ii ◽  
Multi Objective ◽  
Objective Space ◽  
Problem Instances

A number of benchmark problems exist for evaluating multi-objective evolutionary algorithms (MOEAs) in the objective space. However, the decision space performance analysis is a recent and relatively less explored topic in evolutionary multi-objective optimization research. Among other implications, such analysis can lead to designing more realistic test problems, gaining better understanding about optimal and robust design areas, and design and evaluation of knowledge-based optimization algorithms. This paper complements the existing research in this area and proposes a new method to generate multi-objective optimization test problems with clustered Pareto sets in hyper-rectangular defined areas of decision space. The test problem is parametrized to control number of decision variables, number and position of optimal areas in the decision space and modality of fitness landscape. Three leading MOEAs, including NSGA-II, NSGA-III, and MOEA/D, are evaluated on a number of problem instances with varying characteristics. A new metric is proposed that measures the performance of algorithms in terms of their coverage of the optimal areas in the decision space. The empirical analysis presented in this research shows that the decision space performance may not necessarily be reflective of the objective space performance and that all algorithms are sensitive to population size parameter for the new test problems.

Archive of Useful Solutions for Directed Mating in Evolutionary Constrained Multiobjective Optimization

2014 ◽  
Vol 18 (2) ◽  
pp. 221-231 ◽  
Author(s):  
Minami Miyakawa ◽  
◽  
Keiki Takadama ◽  
Hiroyuki Sato
Keyword(s):  
Pareto Front ◽  
Optimization Problems ◽  
Selection Process ◽  
Benchmark Problems ◽  
Search Performance ◽  
Next Generation ◽  
Multi Objective ◽  
Objective Space ◽  
Feasible Solutions ◽  
Constrained Multiobjective Optimization

As an evolutionary approach to solve multi-objective optimization problems involving several constraints, recently a multi-objective evolutionary algorithm (MOEA) using two-stage non-dominated sorting and directed mating (TNSDM) has been proposed. In TNSDM, directed mating utilizes infeasible solutions dominating feasible solutions in the objective space to generate offspring. In our previous studies, significant contribution of directed mating to the improvement of the search performancewas verified on several benchmark problems. However, in the conventional TNSDM, infeasible solutions utilized in directed mating are discarded in the selection process of parents (elites) population and cannot be utilized in the next generation. TNSDM has potential to further improve the search performance by archiving useful solutions for directed mating to the next generation and repeatedly utilizing them in directed mating. To further improve effects of directed mating in TNSDM, in this work, we propose an archiving strategy of useful solutions for directed mating. We verify the search performance of TNSDM using the proposed archive by varying the size of archive, and compare its search performance with the conventional CNSGA-II and RTS onmobjectiveskknapsacks problems. As results, we show that the search performance of TNSDM is improved by introducing the proposed archive in aspects of diversity of the obtained solutions in the objective space and convergence of solutions toward the optimal Pareto front.

scholarly journals Pareto Optimization of a Half Car Passive Suspension Model Using a Novel Multiobjective Heat Transfer Search Algorithm

2017 ◽  
Vol 2017 ◽  
pp. 1-17 ◽  
Author(s):  
Vimal Savsani ◽  
Vivek Patel ◽  
Bhargav Gadhvi ◽  
Mohamed Tawhid
Keyword(s):  
Heat Transfer ◽  
Genetic Algorithm ◽  
Multiobjective Optimization ◽  
Pareto Front ◽  
Search Algorithm ◽  
Extreme Points ◽  
Search Technique ◽  
Nsga Ii ◽  
Linear Ordinary Differential Equations ◽  
Nondominated Sorting

Most of the modern multiobjective optimization algorithms are based on the search technique of genetic algorithms; however the search techniques of other recently developed metaheuristics are emerging topics among researchers. This paper proposes a novel multiobjective optimization algorithm named multiobjective heat transfer search (MOHTS) algorithm, which is based on the search technique of heat transfer search (HTS) algorithm. MOHTS employs the elitist nondominated sorting and crowding distance approach of an elitist based nondominated sorting genetic algorithm-II (NSGA-II) for obtaining different nondomination levels and to preserve the diversity among the optimal set of solutions, respectively. The capability in yielding a Pareto front as close as possible to the true Pareto front of MOHTS has been tested on the multiobjective optimization problem of the vehicle suspension design, which has a set of five second-order linear ordinary differential equations. Half car passive ride model with two different sets of five objectives is employed for optimizing the suspension parameters using MOHTS and NSGA-II. The optimization studies demonstrate that MOHTS achieves the better nondominated Pareto front with the widespread (diveresed) set of optimal solutions as compared to NSGA-II, and further the comparison of the extreme points of the obtained Pareto front reveals the dominance of MOHTS over NSGA-II, multiobjective uniform diversity genetic algorithm (MUGA), and combined PSO-GA based MOEA.

scholarly journals AMOBH: Adaptive Multiobjective Black Hole Algorithm

2017 ◽  
Vol 2017 ◽  
pp. 1-19 ◽  
Author(s):  
Chong Wu ◽  
Tao Wu ◽  
Kaiyuan Fu ◽  
Yuan Zhu ◽  
Yongbo Li ◽  
...  
Keyword(s):  
Black Hole ◽  
Cell Density ◽  
Pareto Front ◽  
Population Diversity ◽  
Evolutionary Strategies ◽  
Nsga Ii ◽  
Strength Assessment ◽  
Objective Space ◽  
Evolution Status ◽  
Black Hole Algorithm

This paper proposes a new multiobjective evolutionary algorithm based on the black hole algorithm with a new individual density assessment (cell density), called “adaptive multiobjective black hole algorithm” (AMOBH). Cell density has the characteristics of low computational complexity and maintains a good balance of convergence and diversity of the Pareto front. The framework of AMOBH can be divided into three steps. Firstly, the Pareto front is mapped to a new objective space called parallel cell coordinate system. Then, to adjust the evolutionary strategies adaptively, Shannon entropy is employed to estimate the evolution status. At last, the cell density is combined with a dominance strength assessment called cell dominance to evaluate the fitness of solutions. Compared with the state-of-the-art methods SPEA-II, PESA-II, NSGA-II, and MOEA/D, experimental results show that AMOBH has a good performance in terms of convergence rate, population diversity, population convergence, subpopulation obtention of different Pareto regions, and time complexity to the latter in most cases.

Adaptive Control of Dominance Area of Solutions in Evolutionary Many-Objective Optimization

2015 ◽  
Vol 11 (02) ◽  
pp. 135-150 ◽  
Author(s):  
Kouhei Tomita ◽  
Minami Miyakawa ◽  
Hiroyuki Sato
Keyword(s):  
Pareto Front ◽  
Optimization Problems ◽  
Benchmark Problems ◽  
Pareto Optimal ◽  
Search Performance ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Pareto Dominance ◽  
Nsga Ii ◽  
Convergence Of Solutions

Controlling the dominance area of solutions (CDAS) relaxes the concept of Pareto dominance with an user-defined parameter S. CDAS with S < 0.5 expands the dominance area and improves the search performance of multi-objective evolutionary algorithms (MOEAs) especially in many-objective optimization problems (MaOPs) by enhancing convergence of solutions toward the optimal Pareto front. However, there is a problem that CDAS with an expanded dominance area (S < 0.5) generally cannot approximate entire Pareto front. To overcome this problem we propose an adaptive CDAS (A-CDAS) that adaptively controls the dominance area of solutions during the solutions search. Our method improves the search performance in MaOPs by approximating the entire Pareto front while keeping high convergence. In early generations, A-CDAS tries to converge solutions toward the optimal Pareto front by using an expanded dominance area with S < 0.5. When we detect convergence of solutions, we gradually increase S and contract the dominance area of solutions to obtain Pareto optimal solutions (POS) covering the entire optimal Pareto front. We verify the effectiveness and the search performance of the proposed A-CDAS on concave and convex DTLZ3 benchmark problems with 2–8 objectives, and show that the proposed A-CDAS achieves higher search performance than conventional non-dominated sorting genetic algorithm II (NSGA-II) and CDAS with an expanded dominance area.

scholarly journals An Improved Multiobjective Algorithm: DNSGA2-PSA

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Dan Qu ◽  
Xianfeng Ding ◽  
Hongmei Wang
Keyword(s):  
Pareto Front ◽  
Optimization Problems ◽  
High Dimensional ◽  
Nsga Ii ◽  
Equal Importance ◽  
Multiobjective Optimization Problems ◽  
Diversity Preservation ◽  
Ranking Scheme ◽  
Nondominated Sorting ◽  
Better Than

In general, the proximities to a certain diversity along the front and the Pareto front have the equal importance for solving multiobjective optimization problems (MOPs). However, most of the existing evolutionary algorithms give priority to the proximity over the diversity. To improve the diversity and decrease execution time of the nondominated sorting genetic algorithm II (NSGA-II), an improved algorithm is presented in this paper, which adopts a new vector ranking scheme to decrease the whole runtime and utilize Part and Select Algorithm (PSA) to maintain the diversity. In this algorithm, a more efficient implementation of nondominated sorting, namely, dominance degree approach for nondominated sorting (DDA-NS), is presented. Moreover, an improved diversity preservation mechanism is proposed to select a well-diversified set out of an arbitrary given set. By embedding PSA and DDA-NS into NSGA-II, denoted as DNSGA2-PSA, the whole runtime of the algorithm is decreased significantly and the exploitation of diversity is enhanced. The computational experiments show that the combination of both (DDA-NS, PSA) to NSGA-II is better than the isolated use cases, and DNSGA2-PSA still performs well in the high-dimensional cases.

A Circle Intersection Method for Bi-Objective Optimization

2021 ◽  
pp. 1-36
Author(s):  
Jianhua Zhou ◽  
Mian Li ◽  
Xiaojin Fu
Keyword(s):  
Pareto Front ◽  
Computational Cost ◽  
Multi Objective Optimization ◽  
Nsga Ii ◽  
New Approach ◽  
Pareto Point ◽  
Objective Space ◽  
Anchor Points ◽  
Intersection Method

Abstract Multi-Objective Optimization (MOO) problems are encountered in many applications, of which bi-objective problems are frequently met. Despite the computational efforts, the quality of the Pareto front is also a considerable issue. An evenly distributed Pareto front is desirable in certain cases when a continuous representation of the Pareto front is needed. In this paper, a new approach called Circle Intersection (CI) is proposed. Firstly, the anchor points are computed. Then in the normalized objective space, a circle with a proper radius of r centering at one of the anchor points or the latest obtained Pareto point is drawn. Interestingly, the intersection of the circle and the feasible boundary can be determined whether it is a Pareto point or not. For a convex or concave feasible boundary, the intersection is exactly the Pareto point, while for other cases the intersection can provide useful information for searching the true Pareto point even if it is not a Pareto point. A novel MOO formulation is proposed for CI correspondingly. Sixteen examples are used to demonstrate the applicability of the proposed method and results are compared to those of NNC, MOGOA, and NSGA-II. Computational results show that the proposed CI method is able to obtain a well-distributed Pareto front with a better quality or with less computational cost.

scholarly journals Constrained Multiobjective Biogeography Optimization Algorithm

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Hongwei Mo ◽  
Zhidan Xu ◽  
Lifang Xu ◽  
Zhou Wu ◽  
Haiping Ma
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Algorithm ◽  
Pareto Front ◽  
Benchmark Problems ◽  
Feasible Region ◽  
Multiple Objective ◽  
Objective Functions ◽  
Nsga Ii ◽  
Constrained Multiobjective Optimization ◽  
Better Than

Multiobjective optimization involves minimizing or maximizing multiple objective functions subject to a set of constraints. In this study, a novel constrained multiobjective biogeography optimization algorithm (CMBOA) is proposed. It is the first biogeography optimization algorithm for constrained multiobjective optimization. In CMBOA, a disturbance migration operator is designed to generate diverse feasible individuals in order to promote the diversity of individuals on Pareto front. Infeasible individuals nearby feasible region are evolved to feasibility by recombining with their nearest nondominated feasible individuals. The convergence of CMBOA is proved by using probability theory. The performance of CMBOA is evaluated on a set of 6 benchmark problems and experimental results show that the CMBOA performs better than or similar to the classical NSGA-II and IS-MOEA.

An Improved Nondominated Sorting Algorithm

2012 ◽  
Vol 3 (4) ◽  
pp. 20-42
Author(s):  
André R. da Cruz
Keyword(s):  
Evolutionary Algorithm ◽  
Pareto Front ◽  
Computational Cost ◽  
Sorting Algorithm ◽  
Nsga Ii ◽  
Worst Case ◽  
New Strategy ◽  
Nondominated Sorting ◽  
Correct Level ◽  
Better Than

This paper presents a new procedure for the nondominated sorting with constraint handling to be used in a multiobjective evolutionary algorithm. The strategy uses a sorting algorithm and binary search to classify the solutions in the correct level of the Pareto front. In a problem with objective functions, using solutions in the population, the original nondominated sorting algorithm, used by NSGA-II, has always a computational cost of in a naïve implementation. The complexity of the new algorithm can vary from in the best case and in the worst case. A experiment was executed in order to compare the new algorithm with the original and another improved version of the Deb’s algorithm. Results reveal that the new strategy is much better than other versions when there are many levels in Pareto front. It is also concluded that is interesting to alternate the new algorithm and the improved Deb’s version during the evolution of the evolutionary algorithm.

scholarly journals Using Different Approaches to Approximate a Pareto Front for a Multiobjective Evolutionary Algorithm: Optimal Thinning Regimes forEucalyptus fastigata

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Oliver Chikumbo
Keyword(s):  
Genetic Algorithm ◽  
Evolutionary Algorithm ◽  
Pareto Front ◽  
Nsga Ii ◽  
Multiobjective Evolutionary Algorithm ◽  
Fitness Sharing ◽  
Pareto Ranking ◽  
Ranking Scheme ◽  
Volume Production ◽  
Nondominated Sorting

A stand-level, multiobjective evolutionary algorithm (MOEA) for determining a set of efficient thinning regimes satisfying two objectives, that is, value production for sawlog harvesting and volume production for a pulpwood market, was successfully demonstrated for aEucalyptus fastigatatrial in Kaingaroa Forest, New Zealand. The MOEA approximated the set of efficient thinning regimes (with a discontinuous Pareto front) by employing a ranking scheme developed by Fonseca and Fleming (1993), which was a Pareto-based ranking (a.k.a Multiobjective Genetic Algorithm—MOGA). In this paper we solve the same problem using an improved version of a fitness sharing Pareto ranking algorithm (a.k.a Nondominated Sorting Genetic Algorithm—NSGA II) originally developed by Srinivas and Deb (1994) and examine the results. Our findings indicate that NSGA II approximates the entire Pareto front whereas MOGA only determines a subdomain of the Pareto points.

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