Multi-Line distance minimization: A visualized many-objective test problem suite

Studying the search behavior of evolutionary many objective optimization is an important, but challenging issue. Existing studies rely mainly on the use of performance indicators which, however, not only encounter increasing difficulties with the number of objectives, but also fail to provide the visual information of the evolutionary search. In this paper, we propose a class of scalable test problems, called multi-line distance minimization problem (ML-DMP), which are used to visually examine the behavior of many-objective search. Two key characteristics of the ML-DMP problem are: 1) its Pareto optimal solutions lie in a regular polygon in the two-dimensional decision space, and 2) these solutions are similar (in the sense of Euclidean geometry) to their images in the high-dimensional objective space. This allows a straightforward understanding of the distribution of the objective vector set (e.g., its uniformity and coverage over the Pareto front) via observing the solution set in the two-dimensional decision space. Fifteen well-established algorithms have been investigated on three types of 10 ML-DMP problem instances. Weakness has been revealed across classic multi-objective algorithms (such as Pareto-based, decomposition based and indicator-based algorithms) and even state-of-the-art algorithms designed especially for many-objective optimization. This, together with some interesting observations from the experimental studies, suggests that the proposed ML-DMP may also be used as a benchmark function to challenge the search ability of optimization algorithms.10.13039/501100000266-Engineering and Physical Sciences Research Council; 10.13039/501100001809-National Natural Science Foundation of China; 10.13039/501100000288-Royal Society

Improving a multi-objective evolutionary algorithm to discover quantitative association rules

This work aims at correcting flaws existing in multi-objective evolutionary schemes to discover quantitative association rules, specifically those based on the wellknown non-dominated sorting genetic algorithm-II (NSGA-II). In particular, a methodology is proposed to find the most suitable configurations based on the set of objectives to optimize and distance measures to rank the non-dominated solutions. First, several quality measures are analyzed to select the best set of them to be optimized. Furthermore, different strate-gies are applied to replace the crowding distance used by NSGA-II to sort the solutions for each Pareto-front since such distance is not suitable for handling many-objective problems. The proposed enhancements have been integrated into the multi-objective algorithm called MOQAR. Several experiments have been carried out to assess the algorithm’s performance by using different configuration settings, and the best ones have been compared to other existing algorithms. The results obtained show a remarkable performance of MOQAR in terms of quality measures.Ministerio de Ciencia y Tecnología TIN2011-28956-C02Ministerio de Ciencia y Tecnología TIN2014- 55894-C2-RJunta de Andalucia P12-TIC-1728Universidad Pablo de Olavide APPB81309

Evolutionary Dynamic Multi-Objective Optimisation : A survey

Peer reviewedPostprin

Diversity comparison of Pareto front approximations in many-objective optimization

Diversity assessment of Pareto front approximations is an important issue in the stochastic multiobjective optimization community. Most of the diversity indicators in the literature were designed to work for any number of objectives of Pareto front approximations in principle, but in practice many of these indicators are infeasible or not workable when the number of objectives is large. In this paper, we propose a diversity comparison indicator (DCI) to assess the diversity of Pareto front approximations in many-objective optimization. DCI evaluates relative quality of different Pareto front approximations rather than provides an absolute measure of distribution for a single approximation. In DCI, all the concerned approximations are put into a grid environment so that there are some hyperboxes containing one or more solutions. The proposed indicator only considers the contribution of different approximations to nonempty hyperboxes. Therefore, the computational cost does not increase exponentially with the number of objectives. In fact, the implementation of DCI is of quadratic time complexity, which is fully independent of the number of divisions used in grid. Systematic experiments are conducted using three groups of artificial Pareto front approximations and seven groups of real Pareto front approximations with different numbers of objectives to verify the effectiveness of DCI. Moreover, a comparison with two diversity indicators used widely in many-objective optimization is made analytically and empirically. Finally, a parametric investigation reveals interesting insights of the division number in grid and also offers some suggested settings to the users with different preferences

Shift-based density estimation for pareto-based algorithms in many-objective optimization

It is commonly accepted that Pareto-based evolutionary multiobjective optimization (EMO) algorithms encounter difficulties in dealing with many-objective problems. In these algorithms, the ineffectiveness of the Pareto dominance relation for a high-dimensional space leads diversity maintenance mechanisms to play the leading role during the evolutionary process, while the preference of diversity maintenance mechanisms for individuals in sparse regions results in the final solutions distributed widely over the objective space but distant from the desired Pareto front. Intuitively, there are two ways to address this problem: 1) modifying the Pareto dominance relation and 2) modifying the diversity maintenance mechanism in the algorithm. In this paper, we focus on the latter and propose a shift-based density estimation (SDE) strategy. The aim of our study is to develop a general modification of density estimation in order to make Pareto-based algorithms suitable for many-objective optimization. In contrast to traditional density estimation that only involves the distribution of individuals in the population, SDE covers both the distribution and convergence information of individuals. The application of SDE in three popular Pareto-based algorithms demonstrates its usefulness in handling many-objective problems. Moreover, an extensive comparison with five state-of-the-art EMO algorithms reveals its competitiveness in balancing convergence and diversity of solutions. These findings not only show that SDE is a good alternative to tackle many-objective problems, but also present a general extension of Pareto-based algorithms in many-objective optimization. © 2013 IEEE

Preventing premature convergence and proving the optimality in evolutionary algorithms

http://ea2013.inria.fr//proceedings.pdfInternational audienceEvolutionary Algorithms (EA) usually carry out an efficient exploration of the search-space, but get often trapped in local minima and do not prove the optimality of the solution. Interval-based techniques, on the other hand, yield a numerical proof of optimality of the solution. However, they may fail to converge within a reasonable time due to their inability to quickly compute a good approximation of the global minimum and their exponential complexity. The contribution of this paper is a hybrid algorithm called Charibde in which a particular EA, Differential Evolution, cooperates with a Branch and Bound algorithm endowed with interval propagation techniques. It prevents premature convergence toward local optima and outperforms both deterministic and stochastic existing approaches. We demonstrate its efficiency on a benchmark of highly multimodal problems, for which we provide previously unknown global minima and certification of optimality