Adriana Lara

Using the Averaged Hausdorff Distance as a Performance Measure in Evolutionary Multiobjective Optimization

The Hausdorff distance dH is a widely used tool to measure the distance between different objects in several research fields. Possible reasons for this might be that it is a natural extension of the well-known and intuitive distance between points and/or the fact that dH defines in certain cases a metric in the mathematical sense. In evolutionary multiobjective optimization (EMO) the task is typically to compute the entire solution set-the so-called Pareto set-respectively its image, the Pareto front. Hence, dH should, at least at first sight, be a natural choice to measure the performance of the outcome set in particular since it is related to the terms spread and convergence as used in EMO literature. However, so far, dH does not find the general approval in the EMO community. The main reason for this is that dH penalizes single outliers of the candidate set which does not comply with the use of stochastic search algorithms such as evolutionary strategies. In this paper, we define a new performance indicator, Δp, which can be viewed as an “averaged Hausdorff distance” between the outcome set and the Pareto front and which is composed of (slight modifications of) the well-known indicators generational distance (GD) and inverted generational distance (IGD). We will discuss theoretical properties of Δp (as well as for GD and IGD) such as the metric properties and the compliance with state-of-theart multiobjective evolutionary algorithms (MOEAs), and will further on demonstrate by empirical results the potential of Δp as a new performance indicator for the evaluation of MOEAs.

HCS: A New Local Search Strategy for Memetic Multiobjective Evolutionary Algorithms

In this paper, we propose and investigate a new local search strategy for multiobjective memetic algorithms. More precisely, we suggest a novel iterative search procedure, known as the Hill Climber with Sidestep (HCS), which is designed for the treatment of multiobjective optimization problems, and show further two possible ways to integrate the HCS into a given evolutionary strategy leading to new memetic (or hybrid) algorithms. The pecularity of the HCS is that it is intended to be capable both moving toward and along the (local) Pareto set depending on the distance of the current iterate toward this set. The local search procedure utilizes the geometry of the directional cones of such optimization problems and works with or without gradient information. Finally, we present some numerical results on some well-known benchmark problems, indicating the strength of the local search strategy as a standalone algorithm as well as its benefit when used within a MOEA. For the latter we use the state of the art algorithms Nondominated Sorting Genetic Algorithm-II and Strength Pareto Evolutionary Algorithm 2 as base MOEAs.

On the Influence of the Number of Objectives on the Hardness of a Multiobjective Optimization Problem

In this paper, we study the influence of the number of objectives of a continuous multiobjective optimization problem on its hardness for evolution strategies which is of particular interest for many-objective optimization problems. To be more precise, we measure the hardness in terms of the evolution (or convergence) of the population toward the set of interest, the Pareto set. Previous related studies consider mainly the number of nondominated individuals within a population which greatly improved the understanding of the problem and has led to possible remedies. However, in certain cases this ansatz is not sophisticated enough to understand all phenomena, and can even be misleading. In this paper, we suggest alternatively to consider the probability to improve the situation of the population which can, to a certain extent, be measured by the sizes of the descent cones. As an example, we make some qualitative considerations on a general class of uni-modal test problems and conjecture that these problems get harder by adding an objective, but that this difference is practically not significant, and we support this by some empirical studies. Further, we address the scalability in the number of objectives observed in the literature. That is, we try to extract the challenges for the treatment of many-objective problems for evolution strategies based on our observations and use them to explain recent advances in this field.

The directed search method for multi-objective memetic algorithms

We propose a new iterative search procedure for the numerical treatment of unconstrained multi-objective optimization problems (MOPs) which steers the search along a predefined direction given in objective space. Based on this idea we will present two methods: directed search (DS) descent which seeks for improvements of the given model, and a novel continuation method (DS continuation) which allows to search along the Pareto set of a given MOP. One advantage of both methods is that they can be realized with and without gradient information, and if neighborhood information is available the computation of the search direction comes even for free. The latter makes our algorithms interesting candidates for local search engines within memetic strategies. Further, the approach can be used to gain some interesting insights into the nature of multi-objective stochastic local search which may explain one facet of the success of multi-objective evolutionary algorithms (MOEAs). Finally, we demonstrate the strength of the method both as standalone algorithm and as local search engine within a MOEA.

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.

Evolutionary continuation methods for optimization problems

In this paper we develop evolutionary strategies for numerical continuation which we apply to scalar and multi-objective optimization problems. To be more precise, we will propose two different methods-an embedding algorithm and a multi-objectivization approach-which are designed to follow an implicitly defined curve where the aim can be to detect the endpoint of the curve (e.g., a root finding problem) or to approximate the entire curve (e.g., the Pareto set of a multi-objective optimization problem). We demonstrate that the novel approaches are very robust in finding the set of interest (point or curve) on several examples.

Using gradient-based information to deal with scalability in multi-objective evolutionary algorithms

This work introduces a hybrid between an elitist multi-objective evolutionary algorithm and a gradient-based descent method, which is applied only to certain (selected) solutions. Our proposed approach requires a low number of objective function evaluations to converge to a few points in the Pareto front. Then, the rest of the Pareto front is reconstructed using a method based on rough sets theory, which also requires a low number of objective function evaluations. Emphasis is placed on the effectiveness of our proposed hybrid approach when increasing the number of decision variables, and a study of the scalability of our approach is also presented.

On Gradient-Based Local Search to Hybridize Multi-objective Evolutionary Algorithms

Using evolutionary algorithms when solving multi-objective optimization problems (MOPs) has shown remarkable results during the last decade. As a consolidated research area it counts with a number of guidelines and processes; even though, their efficiency is still a big issue which lets room for improvements. In this chapter we explore the use of gradient-based information to increase efficiency on evolutionary methods, when dealing with smooth real-valued MOPs. We show the main aspects to be considered when building local search operators using the objective function gradients, and when coupling them with evolutionary algorithms. We present an overview of our current methods with discussion about their convenience for particular kinds of problems.

RDS-NSGA-II: a memetic algorithm for reference point based multi-objective optimization

ABSTRACT Reference point based optimization offers tools for the effective treatment of preference based multi-objective optimization problems, e.g. when the decision-maker has a rough idea about the target objective values. For the numerical solution of such problems, specialized evolutionary strategies have become popular, despite their possible slow convergence rates. Hybridizing such evolutionary algorithms with local search techniques have been shown to produce faster and more reliable algorithms. In this article, the directed search (DS) method is adapted to the context of reference point optimization problems, making this variant, called RDS, a well-suited option for integration into evolutionary algorithms. Numerical results on academic test problems with up to five objectives demonstrate the benefit of the novel hybrid (i.e. the same approximation quality can be obtained more efficiently by the new algorithm), using the state-of-the-art algorithm R-NSGA-II for this coupling. This represents an advantage when treating costly-to-evaluate real-world engineering design problems.

On the detection of nearly optimal solutions in the context of single-objective space mission design problems

When making decisions, having multiple options available for a possible realization of the same project can be advantageous. One way to increase the number of interesting choices is to consider, in addition to the optimal solution x*, also nearly optimal or approximate solutions; these alternative solutions differ from x* and can be in different regions – in the design space – but fulfil certain proximity to its function value f(x*). The scope of this article is the efficient computation and discretization of the set E of ϵ–approximate solutions for scalar optimization problems. To accomplish this task, two strategies to archive and update the data of the search procedure will be suggested and investigated. To make emphasis on data storage efficiency, a way to manage significant and insignificant parameters is also presented. Further on, differential evolution will be used together with the new archivers for the computation of E. Finally, the behaviour of the archiver, as well as the efficiency of the resulting search procedure, will be demonstrated on some academic functions as well as on three models related to space mission design.