On the benefit of ∈-efficient solutions in multi objective space mission design

In this work we consider multi-objective space mission design problems. We will show that it makes sense from the practical point of view to consider in addition to the (Pareto) optimal solutions also nearly optimal ones since this increases significantly the number of options for the decision maker, whereas the possible loss of such approximate solutions compared to optimal - and possibly even 'better' - ones is dispensable. For this, we will examine several typical problems in space trajectory design - a bi-impulsive transfer from the Earth to the asteroid Apophis and several low-thrust multi-gravity assist transfers - and demonstrate the possible benefit of the novel approach. Further, we will present an evolutionary multi-objective algorithm which is designed for this purpose

Approximate solutions in space mission design

In this paper, we address multi-objective space mission design problems. From a practical point of view, it is often the case that,during the preliminary phase of the design of a space mission, the solutions that are actually considered are not 'optimal' (in the Pareto sense)but belong to the basin of attraction of optimal ones (i.e. they are nearly optimal). This choice is motivated either by additional requirements that the decision maker has to take into account or, more often, by robustness considerations. For this, we suggest a novel MOEA which is a modification of the well-known NSGA-II algorithm equipped with a recently proposed archiving strategy which aims at storing the set of approximate solutions of a given MOP. Using this algorithm we will examine some space trajectory design problems and demonstrate the benefit of the novel approach

The Gradient Free Directed Search Method as Local Search within Multi-objective Evolutionary Algorithms

Recently, the Directed Search Method has been proposed as a point-wise iterative search procedure that allows to steer the search, in any direction given in objective space, of a multi-objective optimization problem. While the original version requires the objectives’ gradients, we consider here a possible modification that allows to realize the method without gradient information. This makes the novel algorithm in particular interesting for hybridization with set oriented search procedures, such as multi-objective evolutionary algorithms. In this paper, we propose the DDS, a gradient free Directed Search method, and make a first attempt to demonstrate its benefit, as a local search procedure within a memetic strategy, by integrating the DDS into the well-known algorithmMOEA/D. Numerical results on some benchmark models indicate the advantage of the resulting hybrid

VSD-MOEA: A Dominance-Based Multiobjective Evolutionary Algorithm with Explicit Variable Space Diversity Management

Most state-of-the-art Multiobjective Evolutionary Algorithms (moeas) promote the preservation of diversity of objective function space but neglect the diversity of decision variable space. The aim of this article is to show that explicitly managing the amount of diversity maintained in the decision variable space is useful to increase the quality of moeas when taking into account metrics of the objective space. Our novel Variable Space Diversity-based MOEA (vsd-moea) explicitly considers the diversity of both decision variable and objective function space. This information is used with the aim of properly adapting the balance between exploration and intensification during the optimization process. Particularly, at the initial stages, decisions made by the approach are more biased by the information on the diversity of the variable space, whereas it gradually grants more importance to the diversity of objective function space as the evolution progresses. The latter is achieved through a novel density estimator. The new method is compared with state-of-art moeas using several benchmarks with two and three objectives. This novel proposal yields much better results than state-of-the-art schemes when considering metrics applied on objective function space, exhibiting a more stable and robust behavior

On the Construction of Pareto-Compliant Combined Indicators

The most relevant property that a quality indicator (QI) is expected to have is Pareto compliance, which means that every time an approximation set strictly dominates another in a Pareto sense, the indicator must reflect this. The hypervolume indicator and its variants are the only unary QIs known to be Pareto-compliant but there are many commonly used weakly Pareto-compliant indicators such as R2, IGD+,andɛ+. Currently, an open research area is related to finding new Pareto-compliant indicators whose preferences are different from those of the hypervolume indicator. In this article, we propose a theoretical basis to combine existing weakly Pareto-compliant indicators with at least one being Pareto-compliant, such that the resulting combined indicator is Pareto-compliant as well. Most importantly, we show that the combination of Paretocompliant QIs with weakly Pareto-compliant indicators leads to indicators that inherit properties of the weakly compliant indicators in terms of optimal point distributions. The consequences of these new combined indicators are threefold: (1) to increase the variety of available Pareto-compliant QIs by correcting weakly Pareto-compliant indicators, (2) to introduce a general framework for the combination of QIs, and (3) to generate new selection mechanisms for multiobjective evolutionary algorithms where it is possible to achieve/adjust desired distributions on the Pareto front

Piecewise Linear Representation Segmentation as a Multiobjective Optimization Problem

Proceedings of: Forth International Workshop on User-Centric Technologies and applications (CONTEXTS 2010). Valencia, September 7-10, 2010Actual time series exhibit huge amounts of data which require an unaffordable computational load to be processed, leading to approximate representations to aid these processes. Segmentation processes deal with this issue dividing time series into a certain number of segments and approximating those segments with a basic function. Among the most extended segmentation approaches, piecewise linear representation is highlighted due to its simplicity. This work presents an approach based on the formalization of the segmentation process as a multiobjetive optimization problem and the resolution of that problem with an evolutionary algorithm.This work was supported in part by Projects CICYT TIN2008-06742-C02-02/TSI, CICYT TEC2008-06732-C02-02/TEC, CAM CONTEXTS (S2009/TIC-1485) and DPS2008-07029-C02-02.Publicad

On the Effect of the Cooperation of Indicator-Based Multiobjective Evolutionary Algorithms

For almost 20 years, quality indicators (QIs) have promoted the design of new selection mechanisms of multiobjective evolutionary algorithms (MOEAs). Each indicator-based MOEA (IB-MOEA) has specific search preferences related to its baseline QI, producing Pareto front approximations with different properties. In consequence, an IB-MOEA based on a single QI has a limited scope of multiobjective optimization problems (MOPs) in which it is expected to have a good performance. This issue is emphasized when the associated Pareto front geometries are highly irregular. In order to overcome these issues, we propose here an island-based multiindicator algorithm (IMIA) that takes advantage of the search biases of multiple IB-MOEAs through a cooperative scheme. Our experimental results show that the cooperation of multiple IB-MOEAs allows IMIA to perform more robustly (considering several QIs) than the panmictic versions of its baseline IB-MOEAs as well as several state-of-the-art MOEAs. Additionally, IMIA shows a Pareto-front-shape invariance property, which makes it a remarkable optimizer when tackling MOPs with complex Pareto front geometries

PSA based multi objective evolutionary algorithms

It has generally been acknowledged that both proximity to the Pareto front and a certain diversity along the front, should be targeted when using evolutionary multiobjective optimization. Recently, a new partitioning mechanism, the Part and Select Algorithm (PSA), has been introduced. It was shown that this partitioning allows for the selection of a well-diversified set out of an arbitrary given set, while maintaining low computational cost. When embedded into an evolutionary search (NSGA-II), the PSA has significantly enhanced the exploitation of diversity. In this paper, the ability of the PSA to enhance evolutionary multiobjective algorithms (EMOAs) is further investigated. Two research directions are explored here. The first one deals with the integration of the PSA within an EMOA with a novel strategy. Contrary to most EMOAs, that give a higher priority to proximity over diversity, this new strategy promotes the balance between the two. The suggested algorithm allows some dominated solutions to survive, if they contribute to diversity. It is shown that such an approach substantially reduces the risk of the algorithm to fail in finding the Pareto front. The second research direction explores the use of the PSA as an archiving selection mechanism, to improve the averaged Hausdorff distance obtained by existing EMOAs. It is shown that the integration of the PSA into NSGA-II-I and Δ p -EMOA as an archiving mechanism leads to algorithms that are superior to base EMOAS on problems with disconnected Pareto fronts. © 2014 Springer International Publishing Switzerland

On the utilization of pair-potential energy functions in multi-objective optimization

In evolutionary multi-objective optimization (EMO), the pair-potential energy functions (PPFs) have been used to construct diversity-preserving mechanisms to improve Pareto front approximations. Despite PPFs have shown promising results when dealing with different Pareto front geometries, there are still some open research questions to improve the way we employ them. In this paper, we answer three important questions: (1) what is the effect of a crucial parameter of some PPFs?, (2) how do we set the optimal parameter value?, and (3) what is the best PPF in EMO? To solve these questions, we designed a brand-new fast algorithm to generate an approximate solution to a PPF-based subset selection problem and, then, we conducted a comprehensive parametrical study to predict the optimal parameter values using a deep neural network. To show the effectiveness of the PPF-based diversity-preserving mechanisms, we selected two application cases: the generation of reference point sets of benchmark problems (DTLZ, WFG, IDTLZ, IWFG, IMOP, and Viennet) with different Pareto front shapes, and the definition of a PPF-based archive that can be coupled to any multi-objective evolutionary algorithm to construct well-diversified Pareto front approximations. Using several diversity indicators, it is shown that the utilization of PPF-based mechanisms lead to good Pareto front approximations regardless of the Pareto front shape