A reference point-based evolutionary algorithm for approximating regions of interest in multiobjective problems

Evolutionary algorithms have been widely used to tackle multi-objective optimization problems. Despite many successful applications there is a lack of theoretical knowledge on how and why this type of algorithm works particularly well for multi-objective problems. This special issue presents current advances to the theoretical understanding of evolutionary algorithms for multi-objective optimization. It comprises four very high quality papers which have been selected via a rigorous reviewing process. The first paper “On the Effect of Populations in Evolutionary Multi-Objective Optimisation” by Oliver Giel and Per Kristian Lehre studies the basic question whether a population is provably useful to compute the whole Pareto front of a multi-objective optimization problem. They compare restart strategies of several single-objective approaches based on one individual to a natural multi-objective approach using a population. Their analyses point out the benefits of the multi-objective evolutionary algorithm in a rigorous way. Using illustrative example functions they provide not only proven results but also a good intuition about why population-based approaches can outperform individual-based approaches in multi-objective optimization. The paper “Exploring the Runtime of an Evolutionary Algorithm for the Multi-Objective Shortest Path Problem” by Christian Horoba examines the behavior of evolutionary algorithms for the classical NP-hard multi-objective shortest path problem. The author considers an evolutionary approach based on -dominance and proves that this algorithm constitutes a fully polynomial-time randomized approximation scheme (FPRAS) for the problem. The algorithm achieves this by mimicking a problemspecific algorithm for that problem. This provides general insight into why evolutionary algorithms can perform well for problems that allow for efficient problem-specific algorithms. Hypervolume-based algorithms have become very popular in evolutionary multiobjective optimization. In the paper, “An Efficient Algorithm for Computing Hypervolume Contributions,” Karl Bringmann and Tobias Friedrich present a faster algorithm for determining the individual with the smallest hypervolume contribution in a given population. Furthermore, they study the problem of removing λ > 1 individuals with the smallest hypervolume contribution from a given population and present an elegant and fast method to carry out this task.

In this paper we propose a methodology based on an evolutionary multiobjective optimization algorithm for the problem of up-grading (expansion) the physical topology of transparent optical networks. The aim is to define the new topology layout and the specification of the new optical devices that should be deployed in an existing network in order to improve its performance, considering simultaneously the total installation cost (capital cost) and the total network blocking probability (performance criteria). To accomplish that, we propose a capital cost model for the network. We considered the following physical layer impairments: losses in optical devices, amplified spontaneous emission in optical amplifier and homodyne crosstalk in optical cross connect. The multiobjective optimization evolutionary algorithm used in the optimization process is the NSGA-II. Our proposed methodology can solve the expand physical topology design problem taking into account the physical layer impairments and the capital costs simultaneously. We also present a case study to show the effectiveness of our methodology.

The hypervolume indicator has frequently been used for comparing evolutionary multi-objective optimization (EMO) algorithms. A reference point is needed for hypervolume calculation. However, its specification has not been discussed in detail from a viewpoint of fair performance comparison. A slightly worse point than the nadir point is usually used for hypervolume calculation in the EMO community. In this paper, we propose a reference point specification method for fair performance comparison of EMO algorithms. First, we discuss the relation between the reference point specification and the optimal distribution of solutions for hypervolume maximization. It is demonstrated that the optimal distribution of solutions strongly depends on the location of the reference point when a multi-objective problem has an inverted triangular Pareto front. Next, we propose a reference point specification method based on theoretical discussions on the optimal distribution of solutions. The basic idea is to specify the reference point so that a set of well-distributed solutions over the entire linear Pareto front has a large hypervolume and all solutions in such a solution set have similar hypervolume contributions. Then, we examine whether the proposed method can appropriately specify the reference point through computational experiments on various test problems. Finally, we examine the usefulness of the proposed method in a hypervolume-based EMO algorithm. Our discussions and experimental results clearly show that a slightly worse point than the nadir point is not always appropriate for performance comparison of EMO algorithms.

Polyurethane is used for making mould in soft tooling (ST) process for producing wax/plastic components. These wax components are later used as pattern in investment casting process. Due to low thermal conductivity of polyurethane, cooling time in ST process is long. To reduce the cooling time, thermal conductive fillers are incorporated into polyurethane to make composite mould material. However, addition of fillers affects various properties of the ST process, such as stiffness of the mould box, rendering flow-ability of melt mould material, etc. In the present work, multi-objective optimization of various conflicting objectives (namely maximization of equivalent thermal conductivity, minimization of effective modulus of elasticity, and minimization of equivalent viscosity) of composite material are conducted using evolutionary algorithms (EAs) in order to design particle-reinforced polyurethane composites by finding the optimal values of design parameters. The design parameters include volume fraction of filler content, size and shape factor of filler particle, etc. The Pareto-optimal front is targeted by solving the corresponding multi-objective problem using the NSGA-II procedure. Then, suitable multi-criterion decision-making techniques are employed to select one or a small set of the optimal solution(s) of design parameter(s) based on the higher level information of the ST process for industrial applications. Finally, the experimental study with a typical real industrial application demonstrates that the obtained optimal design parameters significantly reduce the cooling time in soft tooling process keeping other processing advantages.

This study compared the performances of three state-of-theart evolutionary multi-objective optimization algorithms: the Non-Dominated Sorted Genetic Algorithm II (NSGA-II), the Epsilon-Dominance Non-Dominated Sorted Genetic Algorithm II (e -NSGA-II) and the Epsilon-Dominance Multi-Objective Evolutionary Algorithm ( e MOEA), on a constrained biobjective crop-mix planning problem (BCP). The BCP test case objectives were to: (i) maximize crop production and (ii) minimize planting area. The BCP problem was enumerated to provide the true Pareto-optimal solution set to facilitate rigorous testing of the evolutionary multi-objective optimization algorithms. The performances of the three algorithms were assessed and compared using three performance metrics (additive epsilon indicator, inverted generational distance and spacing). For the comparative study, a graphical comparison with respect to true Pareto front of optimal crop planning problem was provided. Results of the analyses indicated that the e MOEA greatly outperforms the NSGAII and the e -NSGA-II.

Evolutionary algorithm has gained a worldwide popularity among multi-objective optimization. The paper proposes a self-adaptive evolutionary algorithm (called SEA) for multi-objective optimization. In the SEA, the probability of crossover and mutation, P cand P m, are varied depending on the fitness values of the solutions. Fitness assignment of SEA realizes the twin goals of maintaining diversity in the population and guiding the population to the true Pareto Front; fitness value of individual not only depends on improved density estimation but also depends on non-dominated rank. The density estimation can keep diversity in all instances including when scalars of all objectives are much different from each other. SEA is compared against the Non-dominated Sorting Genetic Algorithm (NSGA-II) on a set of test problems introduced by the MOEA community. Simulated results show that SEA is as effective as NSGA-II in most of test functions, but when scalar of objectives are much different from each other, SEA has better distribution of non-dominated solutions.

A multi-objective particle swarm optimization (MOPSO) approach for multi-objective economic load dispatch problem in power system is presented in this paper. The economic load dispatch problem is a non-linear constrained multi-objective optimization problem. The proposed MOPSO approach handles the problem as a multi-objective problem with competing and non-commensurable fuel cost, emission and system loss objectives and has a diversity-preserving mechanism using an external memory (call “repository”) and a geographically-based approach to find widely different Pareto-optimal solutions. In addition, fuzzy set theory is employed to extract the best compromise solution. Several optimization runs of the proposed MOPSO approach were carried out on the standard IEEE 30-bus test system. The results revealed the capabilities of the proposed MOPSO approach to generate well-distributed Pareto-optimal non-dominated solutions of multi-objective economic load dispatch. Comparison with Multi-objective Evolutionary Algorithm (MOEA) showed the superiority of the proposed MOPSO approach and confirmed its potential for solving multi-objective economic load dispatch.

This paper investigates a curious case of informed initialization technique to solve difficult multi-objective optimization MOP problems. The initial population was injected with non-exact i.e. approximated nadir objective vectors, which are the boundary solutions of a Pareto optimal front PF. The algorithm then successively improves those boundary solutions and utilizes them to generate non-dominated solutions targeted to the vicinity of the PF along the way. The proposed technique was ported to a standard Evolutionary Multi-objective Optimization EMO algorithm and tested on a wide variety of benchmark MOP problems. The experimental results suggest that the proposed approach is very helpful in achieving extremely fast convergence, especially if an experimenter's goal is to find a set of well distributed trade-off solutions within a fix-budgeted solution evaluations SEs. The proposed approach also ensures a more focused exploration of the underlying search space.

Over the past decades, evolutionary multi-objective optimization algorithms have shown their strength on solving the multi-objective optimization problems. The S-Metric Selection Evolutionary Multiobjective Optimization Algorithm (SMS-EMOA) is a state-of-the-art algorithm which uses the hypervolume indicator as selection criterion and performs well in finding well distributed solutions to approximate the Pareto front. In this paper, the concept of multiple dynamic reference points is integrated to the SMS-EMOA which can balance the trade-off of exploration and exploitation by changing the number of reference points. This way it combines concepts of indicator based and decomposition based EMOA design in a promising manner. The proposed algorithm is compared with other well established EMOA on the classical ZDT benchmark which have 5 different test problems with two objectives. The results show that SMS-EMOA with multiple dynamic reference points outperforms other state-of-the-art algorithms, including SMS-EMOA, by covering more hypervolume which means it has a better approximation of the Pareto front.

Various evolutionary multiobjective optimization (EMO) algorithms have been used in the field of evolutionary fuzzy systems (EFS), because EMO algorithms can easily handle multiple objective functions such as the accuracy maximization and complexity minimization for fuzzy system design. Most EMO algorithms used in EFS are Pareto dominance-based algorithms such as NSGA-II, SPEA2, and PAES. There are a few studies where other types of EMO algorithms are used in EFS. In this paper, we apply a multiobjective evolutionary algorithm based on decomposition called MOEA/D to EFS for fuzzy classifier design. MOEA/D is one of the most well-known decomposition-based EMO algorithms. The key idea is to divide a multiobjective optimization problem into a number of single-objective problems using a set of uniformly distributed weight vectors in a scalarizing function. We propose a new scalarizing function called an accuracy-oriented function (AOF) which is specialized for classifier design. We examine the effects of using AOF in MOEA/D on the search ability of our multiobjective fuzzy genetics-based machine learning (GBML). We also examine the synergy effect of MOEA/D with AOF and parallel distributed implementation of fuzzy GBML on the generalization ability.

The design of Renewable Energy Power Plants (REPPs) is crucial not only for the investments' performance and attractiveness measures, but also for the maximization of resource (source) usage (e.g. sun, water, and wind) and the minimization of raw materials (e.g. aluminum: Al, cadmium: Cd, iron: Fe, silicon: Si, and tellurium: Te) consumption. Hence, several appropriate and satisfactory Multi-Objective Problems (MOPs) are mandatory during the REPPs' design phases. MOPs related tasks can only be managed by very well organized knowledge acquisition on all REPPs' design equations and models. The proposed MOPs need to be solved with one or more multi-objective algorithm, such as Multi-Objective Evolutionary Algorithms (MOEAs). In this respect, the first aim of this research study is to start gathering knowledge on the REPPs' MOPs. The second aim of this study is to gather detailed information about all MOEAs and available free software tools for their development. The main contribution of this research is the initialization of a proposed multi-objective evolutionary algorithm knowledge acquisition system for renewable energy power plants (MOEAs-KAS-F-REPPs) (research and development loopwise process: develop, train, validate, improve, test, improve, operate, and improve). As a simple representative example of this knowledge acquisition system research with two selective and elective proposed standard objectives (as test objectives) and eight selective and elective proposed standard constraints (as test constraints) are generated and applied as a standardized MOP for a virtual small hydropower plant design and investment. The maximization of energy generation (MWh) and the minimization of initial investment cost (million €) are achieved by the Multi-Objective Genetic Algorithm (MOGA), the Niched Sharing Genetic Algorithm/Non-dominated Sorting Genetic Algorithm (NSGA-I), and the NSGA-II algorithms in the Scilab 6.0.0 as only three standardized MOEAs amongst all proposed standardized MOEAs on two desktop computer configurations (Windows 10 Home 1709 64 bits, Intel i5-7200 CPU @ 2.7 GHz, 8.00 GB RAM with internet connection and Windows 10 Pro, Intel(R) Core(TM) i5 CPU 650 @ 3.20 GHz, 6,00 GB RAM with internet connection). The algorithm run-times (computation time) of the current applications vary between 20.64 and 59.98 seconds.

In recent years, multi-objective evolutionary optimization algorithms have shown success in different areas of research. Due to their efficiency and power, many researchers have concentrated on adapting evolutionary algorithms to generate Pareto solutions. This paper proposes a new memetic adaptive multi-objective evolutionary algorithm that is based on a three-term backpropagation network (MAMOT). This algorithm is an automatic search method for optimizing the parameters and performance of neural networks, and it relies on the use of the adaptive non-dominated sorting genetic algorithm-II integrated with the backpropagation algorithm, being used as a local search method. The presented MAMOT employs a self-adaptive mechanism toward improving the performance of the proposed algorithm and a local optimizer improving all the individuals in a population in order to obtain better accuracy and connection weights. In addition, it selects an appropriate number of hidden nodes simultaneously. The proposed method was applied to 11 datasets representing pattern classification problems, including two-class, multi-class and complex data reflecting real problems. Experiments were performed, and the results indicated that the proposed method is viable in pattern classification tasks compared to a multi-objective genetic algorithm based on a three-term backpropagation network (MOGAT) and some of the methods mentioned in the literature. The statistical analysis results of the t test and Wilcoxon signed-ranks test also show that the performance of the proposed method is significantly better than MOGAT.

An improvement decomposition-based multi-objective evolutionary algorithm with uniform design

Multi-objective as well as dynamic characteristics appear in many real-world problems. In order to use multi-objective evolutionary optimization algorithms (MOEA) efficiently, a systematic analysis of the algorithms’ behavior in dynamic environments by means of dynamic test functions is necessary. These functions can be classified into problems with changing Pareto sets and/or Pareto fronts with different dynamic criteria. Thus, a test suite with existing benchmark functions having dynamic behavior and new designed dynamic test functions for yet uncovered cases is proposed. Convergence and solution distribution features of modern MOEA, namely NSGA-II, SPEA2, and MSOPS using different variation operators (Simulated Binary Crossover with Polynomial Mutation and Differential Evolution) will be analyzed. For this reason, a new path integral metric is introduced. Especially the transfer of singleobjective results and the ability of the algorithms to use historically evolved population properties will be discussed.

An innovative method for the automatic design of optical systems is presented and verified. The proposed method is based on a multi-objective evolutionary memetic optimization algorithm. The multi-objective approach simultaneously, but separately, addresses the image quality, tolerance, and complexity of the system. The memetic technique breaks down the search for optical designs in to three different parts or phases: optical glass selection, exploration, and exploitation. The optical glass selection phase defines the most appropriate set of glasses for the system under design. The glass selection phase limits the available glasses from hundreds to just a few, drastically reducing the design space and significantly increasing the efficiency of the automatic design method. The exploration phase is based on an evolutionary algorithm (EA), more specifically, on a problem-tailored generalized extremal optimization (GEO) algorithm, named Optical GEO (O-GEO). The new EA incorporates many features customized for lens design, such as optical system codification and diversity operators. The trade-off systems found in the exploration phase are refined by a local search, based on the damped least square method in the exploitation phase. As a result, the method returns a set of trade-off solutions, generating a Pareto front. Our method delivers alternative and useful insights for the compromise solutions in a lens design problem. The efficiency of the proposed method is verified through real-world examples, showing excellent results for the tested problems.