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

Evolving dynamic multiple-objective optimization problems with objective replacement

This paper studies the strategies for multi-objective optimization in a dynamic environment. In particular, we focus on problems with objective replacement, where some objectives may be replaced with new objectives during evolution. It is shown that the Pareto-optimal sets before and after the objective replacement share some common members. Based on this observation, we suggest the inheritance strategy. When objective replacement occurs, this strategy selects good chromosomes according to the new objective set from the solutions found before objective replacement, and then continues to optimize them via evolution for the new objective set. The experiment results showed that this strategy can help MOGAs achieve better performance than MOGAs without using the inheritance strategy, where the evolution is restarted when objective replacement occurs. More solutions with better quality are found during the same time span

Racing Multi-Objective Selection Probabilities

In the context of Noisy Multi-Objective Optimization, dealing with uncertainties requires the decision maker to define some preferences about how to handle them, through some statistics (e.g., mean, median) to be used to evaluate the qualities of the solutions, and define the corresponding Pareto set. Approximating these statistics requires repeated samplings of the population, drastically increasing the overall computational cost. To tackle this issue, this paper proposes to directly estimate the probability of each individual to be selected, using some Hoeffding races to dynamically assign the estimation budget during the selection step. The proposed racing approach is validated against static budget approaches with NSGA-II on noisy versions of the ZDT benchmark functions

Generalized decomposition and cross entropy methods for many-objective optimization

Decomposition-based algorithms for multi-objective optimization problems have increased in popularity in the past decade. Although their convergence to the Pareto optimal front (PF) is in several instances superior to that of Pareto-based algorithms, the problem of selecting a way to distribute or guide these solutions in a high-dimensional space has not been explored. In this work, we introduce a novel concept which we call generalized decomposition. Generalized decomposition provides a framework with which the decision maker (DM) can guide the underlying evolutionary algorithm toward specific regions of interest or the entire Pareto front with the desired distribution of Pareto optimal solutions. Additionally, it is shown that generalized decomposition simplifies many-objective problems by unifying the three performance objectives of multi-objective evolutionary algorithms – convergence to the PF, evenly distributed Pareto optimal solutions and coverage of the entire front – to only one, that of convergence. A framework, established on generalized decomposition, and an estimation of distribution algorithm (EDA) based on low-order statistics, namely the cross-entropy method (CE), is created to illustrate the benefits of the proposed concept for many objective problems. This choice of EDA also enables the test of the hypothesis that low-order statistics based EDAs can have comparable performance to more elaborate EDAs

A Convergence indicator for Multi-Objective Optimisation Algorithms

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.Comment: Submitted to TEM

ETEA: A euclidean minimum spanning tree-Based evolutionary algorithm for multiobjective optimization

© the Massachusetts Institute of TechnologyAbstract The Euclidean minimum spanning tree (EMST), widely used in a variety of domains, is a minimum spanning tree of a set of points in the space, where the edge weight between each pair of points is their Euclidean distance. Since the generation of an EMST is entirely determined by the Euclidean distance between solutions (points), the properties of EMSTs have a close relation with the distribution and position information of solutions. This paper explores the properties of EMSTs and proposes an EMST-based Evolutionary Algorithm (ETEA) to solve multiobjective optimization problems (MOPs). Unlike most EMO algorithms that focus on the Pareto dominance relation, the proposed algorithm mainly considers distance-based measures to evaluate and compare individuals during the evolutionary search. Specifically in ETEA, four strategies are introduced: 1) An EMST-based crowding distance (ETCD) is presented to estimate the density of individuals in the population; 2) A distance comparison approach incorporating ETCD is used to assign the fitness value for individuals; 3) A fitness adjustment technique is designed to avoid the partial overcrowding in environmental selection; 4) Three diversity indicators-the minimum edge, degree, and ETCD-with regard to EMSTs are applied to determine the survival of individuals in archive truncation. From a series of extensive experiments on 32 test instances with different characteristics, ETEA is found to be competitive against five state-of-the-art algorithms and its predecessor in providing a good balance among convergence, uniformity, and spread.Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom under Grant EP/K001310/1, and the National Natural Science Foundation of China under Grant 61070088