Multiplicative Approximations, Optimal Hypervolume Distributions, and the Choice of the Reference Point

Many optimization problems arising in applications have to consider several objective functions at the same time. Evolutionary algorithms seem to be a very natural choice for dealing with multi-objective problems as the population of such an algorithm can be used to represent the trade-offs with respect to the given objective functions. In this paper, we contribute to the theoretical understanding of evolutionary algorithms for multi-objective problems. We consider indicator-based algorithms whose goal is to maximize the hypervolume for a given problem by distributing {\mu} points on the Pareto front. To gain new theoretical insights into the behavior of hypervolume-based algorithms we compare their optimization goal to the goal of achieving an optimal multiplicative approximation ratio. Our studies are carried out for different Pareto front shapes of bi-objective problems. For the class of linear fronts and a class of convex fronts, we prove that maximizing the hypervolume gives the best possible approximation ratio when assuming that the extreme points have to be included in both distributions of the points on the Pareto front. Furthermore, we investigate the choice of the reference point on the approximation behavior of hypervolume-based approaches and examine Pareto fronts of different shapes by numerical calculations

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

Empirical Investigations of Reference Point Based Methods When Facing a Massively Large Number of Objectives: First Results

EMO 2017: 9th International Conference on Evolutionary Multi-Criterion Optimization, 19-22 March 2017, Münster, GermanyThis is the author accepted manuscript. The final version is available from Springer Verlag via the DOI in this record.Multi-objective optimization with more than three objectives has become one of the most active topics in evolutionary multi-objective optimization (EMO). However, most existing studies limit their experiments up to 15 or 20 objectives, although they claimed to be capable of handling as many objectives as possible. To broaden the insights in the behavior of EMO methods when facing a massively large number of objectives, this paper presents some preliminary empirical investigations on several established scalable benchmark problems with 25, 50, 75 and 100 objectives. In particular, this paper focuses on the behavior of the currently pervasive reference point based EMO methods, although other methods can also be used. The experimental results demonstrate that the reference point based EMO method can be viable for problems with a massively large number of objectives, given an appropriate choice of the distance measure. In addition, sufficient population diversity should be given on each weight vector or a local niche, in order to provide enough selection pressure. To the best of our knowledge, this is the first time an EMO methodology has been considered to solve a massively large number of conflicting objectives.This work was partially supported by EPSRC (Grant No. EP/J017515/1

Using Covariance Matrix Adaptation Evolutionary Strategy to boost the search accuracy in hierarchic memetic computations

Many global optimization problems arising naturally in science and engineering exhibit some form of intrinsic ill-posedness, such as multimodality and insensitivity. Severe ill-posedness precludes the use of standard regularization techniques and necessitates more specialized approaches, usually comprised of two separate stages - global phase, that determines the problem's modality and provides rough approximations of the solutions, and a local phase, which re fines these approximations. In this work, we attempt to improve one of the most efficient currently known approaches - Hierarchic Memetic Strategy (HMS) - by incorporating the Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) into its local phase. CMA-ES is a stochastic optimization algorithm that in some sense mimics the behavior of population-based evolutionary algorithms without explicitly evolving the population. This way, it avoids, to an extent, the associated cost of multiple evaluations of the objective function. We compare the performance of the HMS on relatively simple multimodal benchmark problems and on an engineering problem. To do so, we consider two con gurations: the CMA-ES and the standard SEA (Simple Evolutionary Algorithm). The results demonstrate that the HMS with CMA-ES in the local phase requires less objective function evaluations to provide the same accuracy, making this approach more efficient than the standard SEA

A test problem for visual investigation of high-dimensional multi-objective search

An inherent problem in multiobjective optimization is that the visual observation of solution vectors with four or more objectives is infeasible, which brings major difficulties for algorithmic design, examination, and development. This paper presents a test problem, called the Rectangle problem, to aid the visual investigation of high-dimensional multiobjective search. Key features of the Rectangle problem are that the Pareto optimal solutions 1) lie in a rectangle in the two-variable decision space and 2) are similar (in the sense of Euclidean geometry) to their images in the four-dimensional objective space. In this case, it is easy to examine the behavior of objective vectors in terms of both convergence and diversity, by observing their proximity to the optimal rectangle and their distribution in the rectangle, respectively, in the decision space. Fifteen algorithms are investigated. Underperformance of Pareto-based algorithms as well as most state-of-the-art many-objective algorithms indicates that the proposed problem not only is a good tool to help visually understand the behavior of multiobjective search in a high-dimensional objective space but also can be used as a challenging benchmark function to test algorithms' ability in balancing the convergence and diversity of solutions

Evidence of coevolution in multi-objective evolutionary algorithms

This paper demonstrates that simple yet important characteristics of coevolution can occur in evolutionary algorithms when only a few conditions are met. We find that interaction-based fitness measurements such as fitness (linear) ranking allow for a form of coevolutionary dynamics that is observed when 1) changes are made in what solutions are able to interact during the ranking process and 2) evolution takes place in a multi-objective environment. This research contributes to the study of simulated evolution in a at least two ways. First, it establishes a broader relationship between coevolution and multi-objective optimization than has been previously considered in the literature. Second, it demonstrates that the preconditions for coevolutionary behavior are weaker than previously thought. In particular, our model indicates that direct cooperation or competition between species is not required for coevolution to take place. Moreover, our experiments provide evidence that environmental perturbations can drive coevolutionary processes; a conclusion that mirrors arguments put forth in dual phase evolution theory. In the discussion, we briefly consider how our results may shed light onto this and other recent theories of evolution