Evolutionary many-objective optimisation: pushing the boundaries

This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University LondonMany-objective optimisation poses great challenges to evolutionary algorithms. To start with, the ineffectiveness of the Pareto dominance relation, which is the most important criterion in multi-objective optimisation, results in the underperformance of traditional Pareto-based algorithms. Also, the aggravation of the conflict between proximity and diversity, along with increasing time or space requirement as well as parameter sensitivity, has become key barriers to the design of effective many-objective optimisation algorithms. Furthermore, the infeasibility of solutions' direct observation can lead to serious difficulties in algorithms' performance investigation and comparison. In this thesis, we address these challenges, aiming to make evolutionary algorithms as effective in many-objective optimisation as in two- or three-objective optimisation. First, we significantly enhance Pareto-based algorithms to make them suitable for many-objective optimisation by placing individuals with poor proximity into crowded regions so that these individuals can have a better chance to be eliminated. Second, we propose a grid-based evolutionary algorithm which explores the potential of the grid to deal with many-objective optimisation problems. Third, we present a bi-goal evolution framework that converts many objectives of a given problem into two objectives regarding proximity and diversity, thus creating an optimisation problem in which the objectives are the goals of the search process itself. Fourth, we propose a comprehensive performance indicator to compare evolutionary algorithms in optimisation problems with various Pareto front shapes and any objective dimensionality. Finally, we construct a test problem to aid the visual investigation of evolutionary search, with its Pareto optimal solutions in a two-dimensional decision space having similar distribution to their images in a higher-dimensional objective space. The work reported in this thesis is the outcome of innovative attempts at addressing some of the most challenging problems in evolutionary many-objective optimisation. This research has not only made some of the existing approaches, such as Pareto-based or grid-based algorithms that were traditionally regarded as unsuitable, now effective for many-objective optimisation, but also pushed other important boundaries with novel ideas including bi-goal evolution, a comprehensive performance indicator and a test problem for visual investigation. All the proposed algorithms have been systematically evaluated against existing state of the arts, and some of these algorithms have already been taken up by researchers and practitioners in the field.Department of Computer Science, Brunel University Londo

On the use of two reference points in decomposition based multiobjective evolutionary algorithms

Decomposition based multiobjective evolutionary algorithms approximate the Pareto front of a multiobjective optimization problem by optimizing a set of subproblems in a collaborative manner. Often, each subproblem is associated with a direction vector and a reference point. The settings of these parameters have a very critical impact on convergence and diversity of the algorithm. Some work has been done to study how to set and adjust direction vectors to enhance algorithm performance for particular problems. In contrast, little effort has been made to study how to use reference points for controlling diversity in decomposition based algorithms. In this paper, we first study the impact of the reference point setting on selection in decomposition based algorithms. To balance the diversity and convergence, a new variant of the multiobjective evolutionary algorithm based on decomposition with both the ideal point and the nadir point is then proposed. This new variant also employs an improved global replacement strategy for performance enhancement. Comparison of our proposed algorithm with some other state-of-the-art algorithms is conducted on a set of multiobjective test problems. Experimental results show that our proposed algorithm is promising

Integrating decomposition methods with user preferences for solving many-objective optimization problems

This research aims to investigate methods to solve many-objective (a multi-objective problem with three or more objectives) optimization problems. To achieve this, we propose an algorithm combining user-preference and decomposition approaches. The main reasons that decomposition-based evolutionary multi-objective optimization (EMO) methods are employed in this research are: firstly, they suffer less from the selection pressure issue in comparison to dominance ranking as they rely on decomposition methods such as Weighted-sum, Tchebycheff and Penalty-based Boundary Intersection (PBI) to convert a multi-objective problem into a set of single-objective problems. Secondly, decomposition approaches employ a set of weight vectors which give us a reasonable control of solutions in the objective space. As user-preference approaches alleviate the scalability issue of many-objective problems, they are adopted in this research. User-preference methods can potentially save a considerable amount of computational resources by searching on more desired regions rather than the entire Pareto-optimal front. In this research, user-preference is defined in the form of one or more reference points or directions. The proposed algorithm outperforms R-NSGA-II which is one of popular dominance-based approaches on many-objective optimization problems. Finding a diverse set of solutions is another major challenge for EMOs. The issue of solution diversity is of greater importance when dealing with many-objective problems. In this thesis, we propose an algorithm using a mechanism to update the weight vectors according to feedback that quantifies the uniformity of the solutions in the objective space. Two existing metrics and a newly developed metric are adopted as feedback mechanisms. These metrics allow us to assess the contribution of each solution towards improving the overall uniformity of the solution set in the objective space, and to use this information to update the weight vectors adaptively so that the overall uniformity is maintained. The overarching is to identify sparse areas in the objective space, and move the solutions from the denser to sparse areas. The newly developed metric uses the idea of electrostatic equilibrium to calculate the direction in which each solution should move in order to improve the overall uniformity. As we use decomposition methods in this research, the availability of weight vectors gives us an explicit means of controlling the uniformity of solutions in the objective space. Since existing metrics are neither sufficiently accurate nor scalable to measure the performance of user-preference based EMO algorithms, we develop a new performance metric to fill this gap. The proposed metric uses a composite front as a substitute for the Pareto-optimal front then a preferred region is defined on the composite front. Performances of the new metric are compared against a baseline which relies on knowledge of the Pareto-optimal front. One of the key advantages of the proposed metric is that it does not depend on prior knowledge of the Pareto-optimal front of a particular problem, which is most likely the case in real-world situations

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

Evolutionary Algorithms for Static and Dynamic Multiobjective Optimization

Many real-world optimization problems consist of a number of conflicting objectives that have to be optimized simultaneously. Due to the presence of multiple conflicting ob- jectives, there is no single solution that can optimize all the objectives. Therefore, the resulting multiobjective optimization problems (MOPs) resort to a set of trade-off op- timal solutions, called the Pareto set in the decision space and the Pareto front in the objective space. Traditional optimization methods can at best find one solution in a sin- gle run, thereby making them inefficient to solve MOPs. In contrast, evolutionary algo- rithms (EAs) are able to approximate multiple optimal solutions in a single run. This strength makes EAs good candidates for solving MOPs. Over the past several decades, there have been increasing research interests in developing EAs or improving their perfor- mance, resulting in a large number of contributions towards the applicability of EAs for MOPs. However, the performance of EAs depends largely on the properties of the MOPs in question, e.g., static/dynamic optimization environments, simple/complex Pareto front characteristics, and low/high dimensionality. Different problem properties may pose dis- tinct optimization difficulties to EAs. For example, dynamic (time-varying) MOPs are generally more challenging than static ones to EAs. Therefore, it is not trivial to further study EAs in order to make them widely applicable to MOPs with various optimization scenarios or problem properties. This thesis is devoted to exploring EAs’ ability to solve a variety of MOPs with dif- ferent problem characteristics, attempting to widen EAs’ applicability and enhance their general performance. To start with, decomposition-based EAs are enhanced by incorpo- rating two-phase search and niche-guided solution selection strategies so as to make them suitable for solving MOPs with complex Pareto fronts. Second, new scalarizing functions are proposed and their impacts on evolutionary multiobjective optimization are exten- sively studied. On the basis of the new scalarizing functions, an efficient decomposition- based EA is introduced to deal with a class of hard MOPs. Third, a diversity-first- and-convergence-second sorting method is suggested to handle possible drawbacks of convergence-first based sorting methods. The new sorting method is then combined with strength based fitness assignment, with the aid of reference directions, to optimize MOPs with an increase of objective dimensionality. After that, we study the field of dynamic multiobjective optimization where objective functions and constraints can change over time. A new set of test problems consisting of a wide range of dynamic characteristics is introduced at an attempt to standardize test environments in dynamic multiobjective optimization, thereby aiding fair algorithm comparison and deep performance analysis. Finally, a dynamic EA is developed to tackle dynamic MOPs by exploiting the advan- tages of both generational and steady-state algorithms. All the proposed approaches have been extensively examined against existing state-of-the-art methods, showing fairly good performance in a variety of test scenarios. The research work presented in the thesis is the output of initiative and novel attempts to tackle some challenging issues in evolutionary multiobjective optimization. This re- search has not only extended the applicability of some of the existing approaches, such as decomposition-based or Pareto-based algorithms, for complex or hard MOPs, but also contributed to moving forward research in the field of dynamic multiobjective optimiza- tion with novel ideas including new test suites and novel algorithm design

Effective and Efficient Evolutionary Many-Objective Optimization

This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University LondonMany-objective optimization is core to both artificial intelligence and data analytics as real-world problems commonly involve multiple objectives which are required to be optimized simultaneously. A large number of evolutionary algorithms have been developed to search for a set of Pareto optimal solutions for many-objective optimization problems. It is very rare that a many-objective evolutionary algorithm performs well in terms of both effectiveness and efficiency, two key evaluation criteria. Some algorithms may struggle to guide the solutions towards the Pareto front, e.g., Pareto-based algorithms, while other algorithms may have difficulty in diversifying the solutions evenly over the front on certain problems, e.g., decomposition-based algorithms. Furthermore, some effective algorithms may become very computationally expensive as the number of objectives increases, e.g., indicator-based algorithms. The aim of this thesis is to investigate how to make evolutionary algorithms perform well in terms of effectiveness and efficiency in many-objective optimization. After conducting a review of key concepts and the state of the art in the evolutionary many-objective optimization, this thesis shows how to improve the effectiveness of conventional Pareto-based algorithms on a challenging real-world problem in software engineering. This thesis then explores how to further enhance the effectiveness of leading many-objective evolutionary algorithms in general by extending the capability of a very popular and widely cited bi-goal evolution method. Last but not least, this thesis investigates how to strike a balance between effectiveness and efficiency of evolutionary algorithms when solving many-objective optimization problems. The work reported is based on either real-world or recognized synthetic datasets, and the proposed algorithms are compared and evaluated against leading algorithms in the field. The work does not only demonstrate ways of improving the effectiveness and efficiency of many-objective optimization algorithms but also led to promising areas for future research