Performance Of Multi-objective Evolutionary Algorithms Research Articles

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

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.

Objective reduction based on nonlinear correlation information entropy

Abstract It is hard to obtain the entire solution set of a many-objective optimization problem (MaOP) by multi-objective evolutionary algorithms (MOEAs) because of the difficulties brought by the large number of objectives. However, the redundancy of objectives exists in some problems with correlated objectives (linearly or nonlinearly). Objective reduction can be used to decrease the difficulties of some MaOPs. In this paper, we propose a novel objective reduction approach based on nonlinear correlation information entropy (NCIE). It uses the NCIE matrix to measure the linear and nonlinear correlation between objectives and a simple method to select the most conflicting objectives during the execution of MOEAs. We embed our approach into both Pareto-based and indicator-based MOEAs to analyze the impact of our reduction method on the performance of these algorithms. The results show that our approach significantly improves the performance of Pareto-based MOEAs on both reducible and irreducible MaOPs, but does not much help the performance of indicator-based MOEAs.

Many-objective optimization with corner-based search

The performance of multi-objective evolutionary algorithms can severely deteriorate when applied to problems with 4 or more objectives, called many-objective problems. For Pareto dominance based techniques, available information about some optimal solutions can be used to improve their performance. This is the case of corner solutions. This work considers the behaviour of three multi-objective algorithms [Non-dominated sorting genetic algorithm (NSGA-II), Speed-constrained multi-objective particle swarm optimization (SMPSO) and generalized differential evolution (GDE3)] when corner solutions are inserted into the population at different evolutionary stages. The problem of finding corner solutions is addressed by proposing a new algorithm based in multi-objective particle swarm optimization (MOPSO). Results concerning the behaviour of the aforementioned algorithms in five benchmark problems (DTLZ1-5) and respective analysis are presented.

A directed search strategy for evolutionary dynamic multiobjective optimization

Many real-world multiobjective optimization problems are dynamic, requiring an optimization algorithm that is able to continuously track the moving Pareto front over time. In this paper, we propose a directed search strategy (DSS) consisting of two mechanisms for improving the performance of multiobjective evolutionary algorithms in changing environments. The first mechanism reinitializes the population based on the predicted moving direction as well as the directions that are orthogonal to the moving direction of the Pareto set, when a change is detected. The second mechanism aims to accelerate the convergence by generating solutions in predicted regions of the Pareto set according to the moving direction of the non-dominated solutions between two consecutive generations. The two mechanisms, when combined together, are able to achieve a good balance between exploration and exploitation for evolutionary algorithms to solve dynamic multiobjective optimization problems. We compare DSS with two existing prediction strategies on a variety of test instances having different changing dynamics. Empirical results show that DSS is powerful for evolutionary algorithms to deal with dynamic multiobjective optimization problems.

Performance Metric Ensemble for Multiobjective Evolutionary Algorithms

Evolutionary algorithms have been successfully exploited to solve multiobjective optimization problems. In the literature, a heuristic approach is often taken. For a chosen benchmark problem with specific problem characteristics, the performance of multiobjective evolutionary algorithms (MOEAs) is evaluated via some heuristic chosen performance metrics. The conclusion is then drawn based on statistical findings given the preferable choices of performance metrics. The conclusion, if any, is often indecisive and reveals no insight pertaining to which specific problem characteristics the underlying MOEA could perform the best. In this paper, we introduce an ensemble method to compare MOEAs by combining a number of performance metrics using double elimination tournament selection. The double elimination design allows characteristically poor performance of a quality algorithm to still be able to win it all. Experimental results show that the proposed metric ensemble can provide a more comprehensive comparison among various MOEAs than what could be obtained from a single performance metric alone. The end result is a ranking order among all chosen MOEAs, but not quantifiable measures pertaining to the underlying MOEAs.

The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms

Chaotic maps play an important role in improving evolutionary algorithms (EAs) for avoiding the local optima and speeding up the convergence. However, different chaotic maps in different phases have different effects on EAs. This paper focuses on exploring the effects of chaotic maps and giving comprehensive guidance for improving multiobjective evolutionary algorithms (MOEAs) by series of experiments. NSGA-II algorithm, a representative of MOEAs using the nondominated sorting and elitist strategy, is taken as the framework to study the effect of chaotic maps. Ten chaotic maps are applied in MOEAs in three phases, that is, initial population, crossover, and mutation operator. Multiobjective problems (MOPs) adopted are ZDT series problems to show the generality. Since the scale of some sequences generated by chaotic maps is changed to fit for MOPs, the correctness of scaling transformation of chaotic sequences is proved by measuring the largest Lyapunov exponent. The convergence metricγand diversity metric Δ are chosen to evaluate the performance of new algorithms with chaos. The results of experiments demonstrate that chaotic maps can improve the performance of MOEAs, especially in solving problems with convex and piecewise Pareto front. In addition, cat map has the best performance in solving problems with local optima.

Performance enhancement of multiobjective evolutionary optimisers for truss design using an approximate gradient

This paper proposes hybridisation of evolutionary algorithms (EAs) and an efficient search strategy for truss optimisation. During an optimisation process, function gradients are approximated using already explored design solutions. The approximate gradient is then employed as a local search direction. The approximate gradient operator is integrated into the main search procedure of three multiobjective evolutionary algorithms (MOEAs) leading to three hybrid optimisers. The proposed hybrid strategies along with their original MOEAs are implemented on multiobjective design of truss structures. From the comparative results, it is found that the approximate gradient operator can greatly improve the search performance of MOEAs.

An Indicator for Assessing the Spread of Solutions in Multi-Objective Evolutionary Algorithm

Spread assessment of Pareto set approximations is an important issue in evaluating the performance of multi-objective evolutionary algorithms(MOEAs).This paper proposes a boundary solution set based spread indicator(SI) without the information of the Pareto optimal front.Unlike the existed indicators which only consider extreme solutions,SI estimates the distribution range of nondominated sets by employing boundary solutions.Specifically,this paper gives a detailed investigation of the characteristics and properties of the boundary solution set and discussed the relations and differences between boundary solutions and extreme solutions.Moreover,SI made a distinction of influence on distribution range of different boundary solutions according to their classes and further evaluated the spread of boundary solutions by using hypervolume in low-dimensional space.Additionally,to avoid the impact of different convergence results,SI introduced the centroid of the considered solution sets.From a comparative study on several test problems,SI is found to be available and effective in assessing the spread of MOEAs.

Diversity Management in Evolutionary Many-Objective Optimization

In evolutionary multiobjective optimization, the task of the optimizer is to obtain an accurate and useful approximation of the true Pareto-optimal front. Proximity to the front and diversity of solutions within the approximation set are important requirements. Most established multiobjective evolutionary algorithms (MOEAs) have mechanisms that address these requirements. However, in many-objective optimization, where the number of objectives is greater than 2 or 3, it has been found that these two requirements can conflict with one another, introducing problems such as dominance resistance and speciation. In this paper, two diversity management mechanisms are introduced to investigate their impact on overall solution convergence. They are introduced separately, and in combination, and tested on a set of test functions with an increasing number of objectives (6-20). It is found that the inclusion of one of the mechanisms improves the performance of a well-established MOEA in many-objective optimization problems, in terms of both convergence and diversity. The relevance of this for many-objective MOEAs is discussed.

Application and comparison of hybrid evolutionary multiobjective optimization algorithms for solving task scheduling problem on heterogeneous systems

Task scheduling problem in heterogeneous systems (TSPHS) is a multiobjective optimization problem (MOP). Multiobjective evolutionary algorithms (MOEA) are well suited for solving multiobjective task scheduling problem. In this paper, the two conflicting objectives namely, makespan and reliability are considered. The performance of MOEAs can be improved by hybridization with local search. Hybridization of MOEAs improves the convergence speed to Pareto front. Simple neighborhood search (SNS) algorithm is used as the local search algorithm. The weighted-sum based approach for solving the MOP with its hybrid version is compared. Then the two MOEAs: SPEA2 and NSGA-II are compared with each other in the pure and hybrid version for random task graphs and also for a real-time numerical application graph. The simulations confirm that Hybrid NSGA-II is best suited for solving the task scheduling problem.

Effects of MOEA Temporally Switching Pareto Partial Dominance on Many-objective 0/1 Knapsack Problems

In this work, we propose a novel multi-objective evolutionary algorithm (MOEA) which improves search performance of MOEA especially for many-objective combinatorial optimization problems. Pareto dominance based MOEAs such as NSGA-II and SPEA2 meet difficulty to rank solutions in the population noticeably deteriorating search performance as we increase the number of objectives. In the proposed method, we rank solutions by calculating Pareto partial dominance between solutions using r objective functions selected from m objective functions to induce appropriate selection pressure in many-objective optimization by Pareto-based MOEA. Also, we temporally switch r objective functions among mCr combinations in every interval generations Ig to optimize all of the objective functions throughout the entire evolution process. In this work, we use many-objective 0/1 knapsack problems to show the search performance of the proposed method and analyze its evolution behavior. Simulation results show that there is an optimum value for the number of objective functions r to be considered for the calculation of Pareto partial dominance and the interval (generation numbers) Ig to maximize the entire search performance. Also, the search performance of the proposed method is superior to recent state-of-the-art MOEAs, i.e., IBEA, CDAS and MSOPS. Furthermore, we show that the computational time of the proposed method is much less than IBEA, CDAS and MSOPS, and comparative or sometimes less than NSGA-II.

Local Dominance MOEA Including Control of Dominance Area of Solutions on 0/1 Multiobjective Knapsack Problems

Local dominance has been shown to improve significantly the overall performance of multiobjective evolutionary algorithms (MOEAs) on combinatorial optimization problems. This work proposes the control of dominance area of solutions in local dominance MOEAs to enhance Pareto selection aiming to find solutions with high convergence and diversity properties. We control the expansion or contraction of the dominance area of solutions and analyze its effects on the search performance of a local dominance MOEA using 0/1 multiobjective knapsack problems. We show that convergence can be significantly improved while keeping a good distribution of solutions along the whole true Pareto front by using the local dominance MOEA with expansion of dominance area of solutions. We also show that dominance can be applied within very small neighborhoods by controlling the dominance area of solutions, which reduces significantly the computational cost of the local dominance MOEA.

-Similar Elimination to Enhance Search Performance of Multiobjective Evolutionary Algorithms

In this paper, we propose δ-similar elimination to improve the search performance of multiobjective evolutionary algorithms in combinatorial optimization problems. This method eliminates similar individuals in objective space to fairly distribute selection among the different regions of the instantaneous Pareto front. We investigate four eliminating methods analyzing their effects using NSGA-II. In addition, we compare the search performance of NSGA-II enhanced by our method and NSGA-II enhanced by controlled elitism.

Working principles, behavior, and performance of MOEAs on MNK-landscapes

This work studies the working principles, behavior, and performance of multiobjective evolutionary algorithms (MOEAs) on multiobjective epistatic fitness functions with discrete binary search spaces by using MNK-landscapes. First, we analyze the structure and some of the properties of MNK-landscapes under a multiobjective perspective by using enumeration on small landscapes. Then, we focus on the performance and behavior of MOEAs on large landscapes. We organize our study around selection, drift, mutation, and recombination, the four major and intertwined processes that drive adaptive evolution over fitness landscapes. This work clearly shows pros and cons of the main features of MOEAs, gives a valuable guide for the practitioner on how to set up his/her algorithm, enhance MOEAs, and presents useful insights on how to design more robust and efficient MOEAs.

Enhancing Multiobjective Evolutionary Algorithms by Local Dominance and Local Recombination: Performance Verification in Multiobjective 0/1 Knapsack Problems

This paper proposes a method to enhance single population multiobjective evolutionary algorithms (MOEAs) by searching based on local dominance and local recombination. In this method, first, all fitness vectors of individuals are transformed to polar coordinate vectors in objective function space. Then, the population is iteratively divided into several subpopulations by using declination angles. As a result, each sub-population covers a sub-region in the multiobjective space with its individuals located around the same search direction. Next, local dominance is calculated separately for each sub-population after alignment of its principle search direction by rotation. Selection, recombination, and mutation are applied to individuals within each sub-population. The proposed method can improve the performance of MOEAs that use dominance based selection, and can reduce the entire computational cost to calculate dominance among solutions as well. In this paper we verify the effectiveness of the proposed method obtaining Pareto optimal solutions in two representative MOEAs, i.e. NSGA-II and SPEA2, with Multiobjective 0/1 Knapsack Problems.

Multiobjective optimization and evolutionary algorithms for the application mapping problem in multiprocessor system-on-chip design

Sesame is a software framework that aims at developing a modeling and simulation environment for the efficient design space exploration of heterogeneous embedded systems. Since Sesame recognizes separate application and architecture models within a single system simulation, it needs an explicit mapping step to relate these models for cosimulation. The design tradeoffs during the mapping stage, namely, the processing time, power consumption, and architecture cost, are captured by a multiobjective nonlinear mixed integer program. This paper aims at investigating the performance of multiobjective evolutionary algorithms (MOEAs) on solving large instances of the mapping problem. With two comparative case studies, it is shown that MOEAs provide the designer with a highly accurate set of solutions in a reasonable amount of time. Additionally, analyses for different crossover types, mutation usage, and repair strategies for the purpose of constraints handling are carried out. Finally, a number of multiobjective optimization results are simulated for verification.

Pareto Optimization of Energy Absorption of Square Aluminium Columns Using Multi-Objective Genetic Algorithms

Multi-objective genetic algorithms (GAs) are used for Pareto-approach optimization of the energy absorption of square aluminium columns both with and without aluminium foam filler. In this way, a new diversity-preserving algorithm is proposed to enhance the performance of multi-objective evolutionary algorithms (MOEAs). The important conflicting energy-absorption objectives that have been considered in this work are weight and absorbed energy. The multi-objective optimization is considered for two different cases of energy absorption, namely, non-filled and foam filled. It is also demonstrated that the results of two-objective optimization include those of single-objective optimization and, therefore, provide more choices for optimal design of energy absorption for square aluminium columns with and without aluminium foam filler. Using the obtained Pareto front, it is further found that non-filled columns are preferred over foam-filled ones when the desired energy absorption is no more than 4.8 kJ for square aluminium columns.

Thermodynamic Pareto optimization of turbojet engines using multi-objective genetic algorithms

Multi-objective genetic algorithms (GAs) are used for Pareto approach optimization of thermodynamic cycle of ideal turbojet engines. On this behalf, a new diversity preserving algorithm is proposed to enhance the performance of multi-objective evolutionary algorithms (MOEAs) in optimization problems with more than two objective functions. The important conflicting thermodynamic objectives that have been considered in this work are, namely, specific thrust ( ST), thrust-specific fuel consumption ( TSFC), propulsive efficiency ( η p ), and thermal efficiency ( η t ). In this paper, different pairs of these objective functions have been selected for two-objective optimization processes. Moreover, these objectives have also been considered for a four-objective optimization problem using the new diversity preserving algorithm of this work. The comparison results demonstrate the superiority of the new algorithm in preserving the diversity of non-dominated individuals and the quality of Pareto fronts in both two-objective and 4-objective optimization processes. Further, it is shown that some interesting and important relationships among optimal objective functions and decision variables involved in the thermodynamic cycle of turbojet engines can be discovered consequently. Such important relationships as useful optimal design principles would not have been obtained without the use of a multi-objective optimization approach. It is also demonstrated that the results of four-objective optimization can include those of two-objective optimization and, therefore, provide more choices for optimal design of thermodynamic cycle of ideal turbojet engines.

Random Bit Climbers on Multiobjective MNK-Landscapes: Effiects of Memory and Population Climbing

In this work we give an extension of Kauffman's NK-Landscapes to multiobjective MNK-Landscapes in order to study the effects of epistasis on the performance of multiobjective evolutionary algorithms (MOEAs). This paper focuses on the development of multiobjective random one-bit climbers (moRBCs). We incrementally build several moRBCs and analyze basic working principles of state of the art MOEAs on landscapes of increased epistatic complexity and number of objectives. We specially study the effects of Pareto dominance, non-dominance, and the use of memory and a population to influence the search. We choose an elitist non-dominated sorting multiobjective genetic algorithm (NSGA-II) as a representative of the latest generation of MOEAs and include its results for comparison. We detail the behavior of the climbers and show that population based moRBCs outperform NSGA-II for all values of M and K.

The balance between proximity and diversity in multiobjective evolutionary algorithms

Over the last decade, a variety of evolutionary algorithms (EAs) have been proposed for solving multiobjective optimization problems. Especially more recent multiobjective evolutionary algorithms (MOEAs) have been shown to be efficient and superior to earlier approaches. An important question however is whether we can expect such improvements to converge onto a specific efficient MOEA that behaves best on a large variety of problems. In this paper, we argue that the development of new MOEAs cannot converge onto a single new most efficient MOEA because the performance of MOEAs shows characteristics of multiobjective problems. While we point out the most important aspects for designing competent MOEAs in this paper, we also indicate the inherent multiobjective tradeoff in multiobjective optimization between proximity and diversity preservation. We discuss the impact of this tradeoff on the concepts and design of exploration and exploitation operators. We also present a general framework for competent MOEAs and show how current state-of-the-art MOEAs can be obtained by making choices within this framework. Furthermore, we show an example of how we can separate nondomination selection pressure from diversity preservation selection pressure and discuss the impact of changing the ratio between these components.