Miqing Li

A Grid-Based Evolutionary Algorithm for Many-Objective Optimization

Balancing convergence and diversity plays a key role in evolutionary multiobjective optimization (EMO). Most current EMO algorithms perform well on problems with two or three objectives, but encounter difficulties in their scalability to many-objective optimization. This paper proposes a grid-based evolutionary algorithm (GrEA) to solve many-objective optimization problems. Our aim is to exploit the potential of the grid-based approach to strengthen the selection pressure toward the optimal direction while maintaining an extensive and uniform distribution among solutions. To this end, two concepts-grid dominance and grid difference-are introduced to determine the mutual relationship of individuals in a grid environment. Three grid-based criteria, i.e., grid ranking, grid crowding distance, and grid coordinate point distance, are incorporated into the fitness of individuals to distinguish them in both the mating and environmental selection processes. Moreover, a fitness adjustment strategy is developed by adaptively punishing individuals based on the neighborhood and grid dominance relations in order to avoid partial overcrowding as well as guide the search toward different directions in the archive. Six state-of-the-art EMO algorithms are selected as the peer algorithms to validate GrEA. A series of extensive experiments is conducted on 52 instances of nine test problems taken from three test suites. The experimental results show the effectiveness and competitiveness of the proposed GrEA in balancing convergence and diversity. The solution set obtained by GrEA can achieve a better coverage of the Pareto front than that obtained by other algorithms on most of the tested problems. Additionally, a parametric study reveals interesting insights of the division parameter in a grid and also indicates useful values for problems with different characteristics.

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.

Stable Matching-Based Selection in Evolutionary Multiobjective Optimization

Multiobjective evolutionary algorithm based on decomposition (MOEA/D) decomposes a multiobjective optimization problem into a set of scalar optimization subproblems and optimizes them in a collaborative manner. Subproblems and solutions are two sets of agents that naturally exist in MOEA/D. The selection of promising solutions for subproblems can be regarded as a matching between subproblems and solutions. Stable matching, proposed in economics, can effectively resolve conflicts of interests among selfish agents in the market. In this paper, we advocate the use of a simple and effective stable matching (STM) model to coordinate the selection process in MOEA/D. In this model, subproblem agents can express their preferences over the solution agents, and vice versa. The stable outcome produced by the STM model matches each subproblem with one single solution, and it tradeoffs convergence and diversity of the evolutionary search. Comprehensive experiments have shown the effectiveness and competitiveness of our MOEA/D algorithm with the STM model. We have also demonstrated that user-preference information can be readily used in our proposed algorithm to find a region that decision makers are interested in.

Achieving balance between proximity and diversity in multi-objective evolutionary algorithm

Currently, an alternative framework using the hypervolume indicator to guide the search for elite solutions of a multi-objective problem is studied in the evolutionary multi-objective optimization community very actively, comparing to the traditional Pareto dominance based approach. In this paper, we present a dynamic neighborhood multi-objective evolutionary algorithm based on hypervolume indicator (DNMOEA/HI), which benefits from both Pareto dominance and hypervolume indicator based frameworks. DNMOEA/HI is featured by the employment of hypervolume indicator as a truncation operator to prune the exceeded population, while a well-designed density estimator (i.e., tree neighborhood density) is combined with the Pareto strength value to perform fitness assignment. Moreover, a novel algorithm is proposed to directly evaluate the hypervolume contribution of a single individual. The performance of DNMOEA/HI is verified on a comprehensive benchmark suite, in comparison with six other multi-objective evolutionary algorithms. Experimental results demonstrate the efficiency of our proposed algorithm. Solutions obtained by DNMOEA/HI well approach the Pareto optimal front and are evenly distributed over the front, simultaneously.

Evolutionary Multi-Objective Workflow Scheduling in Cloud

Cloud computing provides promising platforms for executing large applications with enormous computational resources to offer on demand. In a Cloud model, users are charged based on their usage of resources and the required quality of service (QoS) specifications. Although there are many existing workflow scheduling algorithms in traditional distributed or heterogeneous computing environments, they have difficulties in being directly applied to the Cloud environments since Cloud differs from traditional heterogeneous environments by its service-based resource managing method and pay-per-use pricing strategies. In this paper, we highlight such difficulties, and model the workflow scheduling problem which optimizes both makespan and cost as a Multi-objective Optimization Problem (MOP) for the Cloud environments. We propose an evolutionary multi-objective optimization (EMO)-based algorithm to solve this workflow scheduling problem on an infrastructure as a service (IaaS) platform. Novel schemes for problem-specific encoding and population initialization, fitness evaluation and genetic operators are proposed in this algorithm. Extensive experiments on real world workflows and randomly generated workflows show that the schedules produced by our evolutionary algorithm present more stability on most of the workflows with the instance-based IaaS computing and pricing models. The results also show that our algorithm can achieve significantly better solutions than existing state-of-the-art QoS optimization scheduling algorithms in most cases. The conducted experiments are based on the on-demand instance types of Amazon EC2; however, the proposed algorithm are easy to be extended to the resources and pricing models of other IaaS services.

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.

Pareto or Non-Pareto: Bi-Criterion Evolution in Multiobjective Optimization

It is known that Pareto dominance has its own weaknesses as the selection criterion in evolutionary multiobjective optimization. Algorithms based on Pareto criterion (PC) can suffer from problems such as slow convergence to the optimal front and inferior performance on problems with many objectives. Non-Pareto criterion (NPC), such as decomposition-based criterion and indicator-based criterion, has already shown promising results in this regard, but its high selection pressure may lead to the algorithm to prefer some specific areas of the problems Pareto front, especially when the front is highly irregular. In this paper, we propose a bi-criterion evolution (BCE) framework of the PC and NPC, which attempts to make use of their strengths and compensates for each others weaknesses. The proposed framework consists of two parts: PC evolution and NPC evolution. The two parts work collaboratively, with an abundant exchange of information to facilitate each others evolution. Specifically, the NPC evolution leads the PC evolution forward and the PC evolution compensates the possible diversity loss of the NPC evolution. The proposed framework keeps the freedom on the implementation of the NPC evolution part, thus making it applicable for any non-Pareto-based algorithm. In the PC evolution, two operations, population maintenance and individual exploration, are presented. The former is to maintain a set of representative nondominated individuals and the latter is to explore some promising areas that are undeveloped (or not well-developed) in the NPC evolution. Experimental results have shown the effectiveness of the proposed framework. The BCE works well on seven groups of 42 test problems with various characteristics, including those in which Pareto-based algorithms or non-Pareto-based algorithms struggle.

SIP: Optimal Product Selection from Feature Models Using Many-Objective Evolutionary Optimization

A feature model specifies the sets of features that define valid products in a software product line. Recent work has considered the problem of choosing optimal products from a feature model based on a set of user preferences, with this being represented as a many-objective optimization problem. This problem has been found to be difficult for a purely search-based approach, leading to classical many-objective optimization algorithms being enhanced either by adding in a valid product as a seed or by introducing additional mutation and replacement operators that use an SAT solver. In this article, we instead enhance the search in two ways: by providing a novel representation and by optimizing first on the number of constraints that hold and only then on the other objectives. In the evaluation, we also used feature models with realistic attributes, in contrast to previous work that used randomly generated attribute values. The results of experiments were promising, with the proposed (SIP) method returning valid products with six published feature models and a randomly generated feature model with 10,000 features. For the model with 10,000 features, the search took only a few minutes.

Spread Assessment for Evolutionary Multi-Objective Optimization

Convergence, uniformity and spread are three basic issues in comparing the performance of multi-objective evolutionary algorithms. However, most of metrics pay more attention on former two performance indices. In this paper, we introduce a metric for evaluating the spread of non-dominated solutions. Unlike existed metrics only calculating the extreme solutions in objective space, this metric defines boundary concept of non-dominated set. And it evaluates the extent of boundary solutions by projecting them on low-dimensional spaces. Moreover, the centroid of solutions set is introduced to avoid the impact of different convergence result of algorithms. From a comparative study on several test problems, the metric is examined to assess spread of non-dominated solutions set in objective space.

Enhancing diversity for average ranking method in evolutionary many-objective optimization

The average ranking (AR) method has been shown highly effective to provide sufficient selection pressure searching towards Pareto optimal set in many-objective optimization. However, as lack of diversity maintenance mechanism, the obtained final set may only concentrate in a subregion of Pareto front. In this paper, we propose a diversity maintenance strategy for AR to balance convergence and diversity during evolution process. We employ grid to define an adaptive neighborhood for each individual, whose size varies with the number of objectives. Moreover, a layering selection scheme integrates it and AR to pick out well-converged individuals and prohibit or postpone the archive of adjacent individuals. From an extensive comparative study with original AR and two other diversity maintenance methods, the proposed method shows a good balance among convergence, uniformity and spread.