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

On the Impact of Multiobjective Scalarizing Functions

Recently, there has been a renewed interest in decomposition-based approaches for evolutionary multiobjective optimization. However, the impact of the choice of the underlying scalarizing function(s) is still far from being well understood. In this paper, we investigate the behavior of different scalarizing functions and their parameters. We thereby abstract firstly from any specific algorithm and only consider the difficulty of the single scalarized problems in terms of the search ability of a (1+lambda)-EA on biobjective NK-landscapes. Secondly, combining the outcomes of independent single-objective runs allows for more general statements on set-based performance measures. Finally, we investigate the correlation between the opening angle of the scalarizing function's underlying contour lines and the position of the final solution in the objective space. Our analysis is of fundamental nature and sheds more light on the key characteristics of multiobjective scalarizing functions.Comment: appears in Parallel Problem Solving from Nature - PPSN XIII, Ljubljana : Slovenia (2014

An adaptation reference-point-based multiobjective evolutionary algorithm

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.It is well known that maintaining a good balance between convergence and diversity is crucial to the performance of multiobjective optimization algorithms (MOEAs). However, the Pareto front (PF) of multiobjective optimization problems (MOPs) affects the performance of MOEAs, especially reference point-based ones. This paper proposes a reference-point-based adaptive method to study the PF of MOPs according to the candidate solutions of the population. In addition, the proportion and angle function presented selects elites during environmental selection. Compared with five state-of-the-art MOEAs, the proposed algorithm shows highly competitive effectiveness on MOPs with six complex characteristics