Using the Averaged Hausdorff Distance as a Performance Measure in Evolutionary Multiobjective Optimization

So far, many performance indicators have been proposed to compare different evolutionary multiobjective optimization algorithms (MOEAs). Among them, the inverted generational distance (IGD) is one of the most commonly used, mainly because it can measure a population’s convergence, diversity, and evenness. However, the effectiveness of IGD highly depends on the quality of the reference set. That is to say, all the reference points should be as close to the Pareto front (PF) as possible and evenly distributed to become ready for a fair performance evaluation. Currently, it is still challenging to generate well-configured reference sets, even if the PF can be given analytically. Therefore, biased reference sets might be a significant source of systematic error. However, in most MOEA literature, biased reference sets are utilized in experiments without an error estimation, which may make the experimental results unconvincing. In this paper, we propose an exact IGD (eIGD) for continuous PF, which is derived from the original IGD under an additional assumption that the reference set is perfect, i.e., the PF itself is directly utilized as an infinite-sized reference set. Therefore, the IGD values produced by biased reference sets can be compared with eIGD so that systematic error can be quantitatively evaluated and analyzed.KeywordsMultiobjective optimizationEvolutionary computationPerformance indicatorReference setDifferential geometry

The problem of atmospheric duct inversion is usually solved as a single objective optimization problem. Based on ground-based Global Positioning System (GPS) phase delay and propagation loss, this paper develops a multi-objective method including the effect of source frequency and receiving antenna height. The diversity and convergence of solution sets are evaluated for seven multi-objective evolutionary algorithms with three performance metrics: Hypervolume (HV), Inverted Generational Distance (IGD), and the averaged Hausdorff distance (). The inversion results are compared with the simulation results, and the experimental comparison is conducted on three groups of test situations. The results demonstrate that the ranking of algorithm performance varies because of the different methods used to calculate performance metrics. Moreover, when the algorithms show overwhelming performance using performance metrics, the inversion result is not more close to the real value. In the comparison of computational experiments, it was found that, as the retrieved parameter dimension increases, the inversion result becomes more unstable. When the observed data are sufficient, the inversion result seems to be improved.

The hypervolume and the inverted generational distance (IGD) have been frequently used for the comparison of evolutionary multiobjective optimization algorithms. For the hypervolume, the relation between the location of a reference point and the optimal distribution of solutions has been studied in the literature. However, such a relation has not been studied for the IGD whereas IGD-based comparison results depend on the specification of reference points. Our intention is to clearly demonstrate the dependency of IGD-based comparison results on reference point specification. First, we explain difficulties of fair comparison in the following two cases: one is the use of all nondominated solutions among obtained solutions by compared algorithms as reference points, and the other is the use of a small number of uniformly sampled reference points. Discussions on these two cases show the necessity of a large number of uniformly sampled reference points on the entire Pareto front. Then, we show a bias of the IGD with such a reference point set through computational experiments. It is shown that the IGD tends to favor a solution set with much smaller diversity than a fully expanded solution set over the entire Pareto front. Finally, we propose a new specification method where reference points are uniformly sampled not only from the Pareto front but also from outside the Pareto front.

The algorithms of multi-objective optimisation had a relative growth in the last years. Thereby, it's requires some way of comparing the results of these. In this sense, performance measures play a key role. In general, it's considered some properties of these algorithms such as capacity, convergence, diversity or convergence-diversity. There are some known measures such as generational distance (GD), inverted generational distance (IGD), hypervolume (HV), Spread($\Delta$), Averaged Hausdorff distance ($\Delta_p$), R2-indicator, among others. In this paper, we focuses on proposing a new indicator to measure convergence based on the traditional formula for Shannon entropy. The main features about this measure are: 1) It does not require tho know the true Pareto set and 2) Medium computational cost when compared with Hypervolume.

The algorithms of multi-objective optimisation had a relative growth in the last years. Thereby, it's requires some way of comparing the results of these. In this sense, performance measures play a key role. In general, it's considered some properties of these algorithms such as capacity, convergence, diversity or convergence-diversity. There are some known measures such as generational distance (GD), inverted generational distance (IGD), hypervolume (HV), Spread($\Delta$), Averaged Hausdorff distance ($\Delta_p$), R2-indicator, among others. In this paper, we focuses on proposing a new indicator to measure convergence based on the traditional formula for Shannon entropy. The main features about this measure are: 1) It does not require tho know the true Pareto set and 2) Medium computational cost when compared with Hypervolume.

In evolutionary multi-objective optimization, it has been illuminated that guide search with neighboring solutions improve the quality of new trial solutions and accelerate algorithms convergence by the regularity property of the continuous multi-objective optimization problems (MOPs). Very recently, clustering learning-based mating strategies have been popular for establishing reproduction operators with neighboring solutions. However, the existing mating strategies may be more reasonable with the full consideration and utilization of the regularity property. The current mating restrictions excessively emphasize algorithm convergence and ignore population diversity. In addition, the selected clustering algorithms in mating restrictions are not conducive for solving MOPs, which have complex Pareto sets (PSs) and\or Pareto fronts (PFs). To solve above problems and address both the algorithm convergence and the population diversity of multi-objective evolutionary algorithms (MOEAs), the spectral clustering based multi-source mating selection strategy (SMMS) is designed to detect regularity properties and balance population diversity while accelerating algorithm convergence. Consequently, a spectral clustering based multi-source mating selection multi-objective evolutionary algorithms is proposed, teamed SMMEA. SMMEA is applied to a number of test suites with a complex PS or PF, and compared with six state-of-the-art MOEAs. The results demonstrate that the proposed algorithm outperforms over the other approaches.

Assessing the performance of Pareto front (PF) approximations is a key issue in the field of evolutionary multi/many-objective optimization. Inverted generational distance (IGD) has been widely accepted as a performance indicator for evaluating the comprehensive quality for a PF approximation. However, IGD usually becomes infeasible when facing a real-world optimization problem as it needs to know the true PF a priori. In addition, the time complexity of IGD grows quadratically with the size of the solution/reference set. To address the aforementioned issues, a grid-based IGD (Grid-IGD) is proposed to estimate both convergence and diversity of PF approximations for multi/many-objective optimization. In Grid-IGD, a set of reference points is generated by estimating PFs of the problem in question, based on the representative nondominated solutions of all the approximations in a grid environment. To reduce the time complexity, Grid-IGD only considers the closest solution within the grid neighborhood in the approximation for every reference point. Grid-IGD also possesses other desirable properties, such as Pareto compliance, immunity to dominated/duplicate solutions, and no need of normalization. In the experimental studies, Grid-IGD is verified on both the artificial and real PF approximations obtained by five many-objective optimizers. Effects of the grid specification on the behavior of Grid-IGD are also discussed in detail theoretically and experimentally.

Runtime analysis of a multi-objective evolutionary algorithm for obtaining finite approximations of Pareto fronts

Evolutionary Multi-objective Optimization(EMO) is becoming a hot research area and quite a few aspects of Multi-objective Evolutionary algorithms(MOEAs) have been studied and discussed. However there are still few literatures discussing the roles of search and selection operators in MOEAs. This paper studied their roles by a representative combinatorial Multi-objective Problem(MOP): Multi-objective TSP. In the new MOEA, We adopt an efficient search operator, which has the properties of both crossover and mutation, to generate the new individuals and chose two kinds of selection operators: Family Competition and Population Competition with probabilities to realize selection. The simulation experiments showed that this new MOEA could get good uniform solutions representing the Pareto Front and outperformed SPEA in almost every simulation run on this problem. Furthermore, we analyzed its convergence property using Finite Markov Chain and proved that it could converge to Pareto Front with probability 1.We also find that the convergence property of MOEAs has much relationship with search and selection operators.

An Area Partitioning and Allocation (APA) approach was presented in[1]. The approach focused on optimizing the coverage performance of Autonomous Industrial Robots (AIRs) using multiple conflicting objectives and Voronoi partitioning. However, questions related to the optimality, convergence, and consistency of the Pareto solutions were not studied in details. In this paper, Inverted Generational Distance (IGD) metric is used to verify the convergence of the Pareto front towards Pareto optimal front (PF*). The consistency in obtaining similar Pareto fronts for independent optimization runs is studied. The computational complexity of the approach with respect to the size of the coverage area and the number of AIRs is also discussed. Two application scenarios are used in this research.

Testing and assessing the performance of proposed algorithms in a right way is a very important part in evolutionary optimization. If the number of optimization objectives is more than three, it's commonly referred as many-objective optimization problem(MaOP). Despite the fact there exists many test functions and assessment metrics, all of which are designed for testing and assessing the performance of evolutionary algorithms, the test functions DTLZ proposed by Deb(2005) and the assessment metrics Inverted Generational Distance(IGD) proposed by Bosman and Thierens(2003) are both widely recognized and used in most of evolutionary optimizations, especially in many-objective evolutionary optimization. When using IGD as a assessment tool, the vital task is to distribute reference points uniformly along the Pareto front of test functions. To overcome it, however, is really difficult with the number of objectives increase. For dealing with such difficulty, this paper introduce an approach called Inverse Transform Technique(ITT) suggested by Fang and Wang(1999) and successfully distributes reference points uniformly along the Pareto front of DTLZ test functions. In the final, three representative evolutionary algorithms: NSGA-II, MOEA/D and MOEA/D-M2M are selected to test their performance in dealing with many-objective optimization. As a result, extensive experiments and analysis are conducted on four DTLZ test functions from five to ten objectives.

FP-ABC: Fuzzy-Pareto dominance driven artificial bee colony algorithm for many-objective software module clustering

In the field of single-objective optimization, hybrid variants of gradient based methods and evolutionary algorithms have been shown to performance better than the pure evolutionary method. This same idea has been used with Evolutionary Multiobjective Optimization (EMO), obtaining also very promising results. In most of the cases, gradient information is used as part of the mutation operator, in order to move every generated point to the exact Pareto front. This means that gradient information is used along the whole process, and then consumes computational resources also along the whole process. On the other hand, in our approach we will use gradient information only at the beginning of the process, and will show that quality of the results is not decreased while computational cost is. We will use a steepest descent method to generate some efficient points to be used to seed an EMO method. The main goal will be generating some efficient points in the exact front using the less evaluations as possible, and let the EMO method use these points to spread along the whole Pareto front. In our approach, we will solve box-constrained continuous problems, gradients will be approximated using quadratic regressions and the EMO method will be based on Rough Sets theory Hernandez-Diaz et al. (Parallel Problem Solving from Nature (PPSN IX) 9th International Conference, 2006).

In a wide range of engineering applications, complex constrained multi-objective optimization problems (CMOPs) present significant challenges, as the complexity of constraints often hampers algorithmic convergence and reduces population diversity. To address these challenges, we propose a novel algorithm named Constraint Intensity-Driven Evolutionary Multitasking (CIDEMT), which employs a two-stage, tri-task framework to dynamically integrates problem structure and knowledge transfer. In the first stage, three cooperative tasks are designed to explore the Constrained Pareto Front (CPF), the Unconstrained Pareto Front (UPF), and the -relaxed constraint boundary, respectively. A CPF-UPF relationship classifier is employed to construct a problem-type-aware evolutionary strategy pool. At the end of the first stage, each task selects strategies from this strategy pool based on the specific type of problem, thereby guiding the subsequent evolutionary process. In the second stage, while each task continues to evolve, a -driven knowledge transfer mechanism is introduced to selectively incorporate effective solutions across tasks. enhancing the convergence and feasibility of the main task. Extensive experiments conducted on 32 benchmark problems from three test suites (LIRCMOP, DASCMOP, and DOC) demonstrate that CIDEMT achieves the best Inverted Generational Distance (IGD) values on 24 problems and the best Hypervolume values (HV) on 22 problems. Furthermore, CIDEMT significantly outperforms six state-of-the-art constrained multi-objective evolutionary algorithms (CMOEAs). These results confirm CIDEMT’s superiority in promoting convergence, diversity, and robustness in solving complex CMOPs.

Archival of elite solutions is widespread practice in evolutionary multi-objective optimization. Grid-based archiving presents a compromise between accuracy and computational cost. Most grid-based archiving algorithms require apriori knowledge of the span of the Pareto front for pre-setting of the grid length or the associated parameter, grid number. Unfortunately the knowledge is often unavailable beforehand in practice. The quality of the attained non-dominated front can be very sensitive to the dimension of the grids. This paper presents a dynamic grid resizing strategy, capable of shrinking or expanding hyper grids as necessity dictates. Empirical study on two- and three-objective test functions demonstrates robust performance with respect to the initial grid sizes. Applied in the context of PAES, the adaptive archiving strategy performed well for initial grid sizes determined from a uniform random distribution. In comparison to AGA, the dynamic strategy presents improved non-dominated solutions in terms of proximity to the Pareto front and diversity for selected test problems.