Optimal μ-Distributions for the Hypervolume Indicator for Problems With Linear Bi-Objective Fronts: Exact and Exhaustive Results

corrected author versionInternational audienceTo simultaneously optimize multiple objective functions, several evolutionary multiobjective optimization (EMO) algorithms have been proposed. Nowadays, often set quality indicators are used when comparing the performance of those algorithms or when selecting ``good'' solutions during the algorithm run. Hence, characterizing the solution sets that maximize a certain indicator is crucial---complying with the optimization goal of many indicator-based EMO algorithms. If these optimal solution sets are upper bounded in size, e.g., by the population size μ, we call them optimal μ-distributions. Recently, optimal μ-distributions for the well-known hypervolume indicator have been theoretically analyzed, in particular, for bi-objective problems with a linear Pareto front. Although the exact optimal μ-distributions have been characterized in this case, not all possible choices of the hypervolume's reference point have been investigated. In this paper, we revisit the previous results and rigorously characterize the optimal μ-distributions also for all other reference point choices. In this sense, our characterization is now exhaustive as the result holds for any linear Pareto front and for any choice of the reference point and the optimal μ-distributions turn out to be always unique in those cases. We also prove a tight lower bound (depending on μ) such that choosing the reference point above this bound ensures the extremes of the Pareto front to be always included in optimal μ-distributions

Comparing Mirrored Mutations and Active Covariance Matrix Adaptation in the IPOP-CMA-ES on the Noiseless BBOB Testbed

International audienceThis paper investigates two variants of the well-known Covariance Matrix Adaptation Evolution Strategy (CMA-ES). Active covariance matrix adaptation allows for negative weights in the covariance matrix update rule such that "bad" steps are (actively) taken into account when updating the covariance matrix of the sample distribution. On the other hand, mirrored mutations via selective mirroring also take the "bad" steps into account. In this case, they are first evaluated when taken in the opposite direction (mirrored) and then considered for regular selection. In this study, we investigate the difference between the performance of the two variants empirically on the noiseless BBOB testbed. The CMA-ES with selectively mirrored mutations only outperforms the active CMA-ES on the sphere function while the active variant statistically significantly outperforms mirrored mutations on 10 of 24 functions in several dimensions

On the Effect of Mirroring in the IPOP Active CMA-ES on the Noiseless BBOB Testbed

International audienceMirrored mutations and active covariance matrix adaptation are two recent ideas to improve the well-known covariance matrix adaptation evolution strategy (CMA-ES)---a state-of-the-art algorithm for numerical optimization. It turns out that both mechanisms can be implemented simultaneously. In this paper, we investigate the impact of mirrored mutations on the so-called IPOP active CMA-ES. We find that additional mirrored mutations improve the IPOP active CMA-ES statistically significantly, but by only a small margin, on several functions while never a statistically significant performance decline can be observed. Furthermore, experiments on different function instances with some algorithm parameters and stopping criteria changed reveal essentially the same results

Interactive Optimization With Weighted Hypervolume Based EMO Algorithms: Preliminary Experiments

The objective functions in multiobjective optimization problems are often non-linear, noisy, or not available in a closed form and evolutionary multiobjective optimization (EMO) algorithms have been shown to be well applicable in this case. Nowadays, for example within the scope of sustainable development, many objectives are taken into account: besides classical objectives such as cost and profit, some new objectives like energy consumption, noise levels or risks have to be considered. With more and more objectives, the number of incomparable alternatives typically increases and the complexity of these problems does not make it easy for a decision maker to formalize preferences towards a specific solution or not even towards a specific but small enough portion of the search space. Moreover, also the algorithms themselves have difficulties to find a good approximation of the entire Pareto front if the number of incomparable solutions increases and the Pareto dominance relation does not indicate a good search direction anymore. In this case, combining the decision making with the search algorithm to an interactive optimization algorithm is considered as a valuable approach. While better and better solutions are found by the optimization algorithm, the DM can specify the preferences more and more precisely while learning about the problem and the objectives' inherent tradeoffs. Such an interactive approach should profit from evaluating solutions only within the interesting regions of the search space in terms of a faster convergence towards the DM's preferred solutions. In the field of EMO, interactive optimization has only been considered recently and in comparison to the vast amount of general EMO algorithms, significantly less interactive EMO algorithms exist. Although, for example, optimization algorithms based on the weighted hypervolume indicator allow to incorporate various preference types into the search, no effort has been made to use this concept within an interactive algorithm. In this report, we propose and discuss how to combine interactive decision making and weighted hypervolume based search algorithms. We focus on a basic model where the DM is asked to pick the most desirable solution among a set. Several examples on standard test problems show the working principles and the usefulness of the interactive approach, in particular with respect to the proximity of the algorithm's population to the DM's most preferred solution.Les fonctions objectif en optimisation multi-objectif sont souvent non-linéaires, bruitées ou non-disponibles et l'optimisation multi-objectif évolutionnaire est applicable dans ce cas. De nos jours, par exemple dans le développement durable, plusieurs objectifs peuvent être pris en compte : en plus des objectifs "classiques" comme le coût et le profit, de "nouveaux" objectifs comme consommation d'énergie, niveaux de bruits ou de risque sont considérés. Avec de plus en plus d'objectifs à prendre en compte, le nombre d'alternatives incomparables croit exponentiellement et la complexité de ces problèmes ne permet pas aux décideurs de formaliser ses préférences afin de calculer une solution spécifique ou même restreindre la recherche à un petit ensemble d'alternatives. De plus, les algorithmes ont des difficultés à trouver une bonne approximation de la région Pareto si le nombre d'alternatives incomparables est grand et la relation de dominance de Pareto ne permet plus une bonne direction de la recherche. Dans ce cas, combiner les algorithmes de recherche et la prise de décision en un algorithme d'optimisation interactif est considérée comme une approche alternative. Pendant que de meilleures solutions sont trouvées par l'algorithme d'optimisation, le décideur peut spécifier ses préférences de manière de plus en plus spécifique en apprenant le problème et le compromis entre les objectifs. Une telle approche interactive devrait bénéficier de l'évaluation des solutions seulement dans des régions intéressantes de l'espace de recherche en terme d'une convergence plus rapide vers les solutions préférées pour le décideur. Dans le domaine de l'optimisation multi-objectif évolutionnaire, l'optimisation interactive a été seulement considérée récemment et en comparaison au grand nombre algorithmes d'optimisation multi-objectif évolutionnaire, peu d'algorithmes d'optimisation multi-objectif évolutionnaire interactifs existent. Bien que, par exemple, des algorithmes d'optimisation basés sur l'indicateur d'hyper-volume pondéré permettent d'inclure plusieurs types de préférences dans la recherche, aucun effort n'a été fourni pour utiliser ce concept dans les algorithmes interactifs. Dans ce rapport, nous proposons et discutons comment combiner la prise de décision interactive et les algorithmes de recherche basés sur l'hyper-volume pondéré. Nous considérons le modèle basique où le décideur est appelé à choisir les solutions qu'il préfère dans un ensemble de solutions. Plusieurs exemples de problèmes de tests standards montrent les principes et l'intérêt de l'approche interactive, en particulier par rapport à la proximité de la population de l'algorithme aux solutions préférées du décideur

COCO: Performance Assessment

We present an any-time performance assessment for benchmarking numerical optimization algorithms in a black-box scenario, applied within the COCO benchmarking platform. The performance assessment is based on runtimes measured in number of objective function evaluations to reach one or several quality indicator target values. We argue that runtime is the only available measure with a generic, meaningful, and quantitative interpretation. We discuss the choice of the target values, runlength-based targets, and the aggregation of results by using simulated restarts, averages, and empirical distribution functions

COCO: The Experimental Procedure

We present a budget-free experimental setup and procedure for benchmarking numericaloptimization algorithms in a black-box scenario. This procedure can be applied with the COCO benchmarking platform. We describe initialization of and input to the algorithm and touch upon therelevance of termination and restarts.Comment: ArXiv e-prints, arXiv:1603.0877

Benchmarking the Local Metamodel CMA-ES on the Noiseless BBOB'2013 Test Bed

International audienceThis paper evaluates the performance of a variant of the local-meta-model CMA-ES (lmm-CMA) in the BBOB 2013 expensive setting. The lmm-CMA is a surrogate variant of the CMA-ES algorithm. Function evaluations are saved by building, with weighted regression, full quadratic metamodels to estimate the candidate solutions' function values. The quality of the approximation is appraised by checking how much the predicted rank changes when evaluating a fraction of the candidate solutions on the original objective function. The results are compared with the CMA-ES without meta-modeling and with previously benchmarked algorithms, namely BFGS, NEWUOA and saACM. It turns out that the additional meta-modeling improves the performance of CMA-ES on almost all BBOB functions while giving significantly worse results only on the attractive sector function. Over all functions, the performance is comparable with saACM and the lmm-CMA often outperforms NEWUOA and BFGS starting from about 2D^2 function evaluations with D being the search space dimension

On the Impact of Active Covariance Matrix Adaptation in the CMA-ES With Mirrored Mutations and Small Initial Population Size on the Noiseless BBOB Testbed

International audienceMirrored mutations as well as active covariance matrix adaptation are two techniques that have been introduced into the well-known CMA-ES algorithm for numerical optimization. Here, we investigate the impact of active covariance matrix adaptation in the IPOP-CMA-ES with mirrored mutation and a small initial population size. Active covariance matrix adaptation improves the performance on 8 of the 24 benchmark functions of the noiseless BBOB test bed. The effect is the largest on the ill-conditioned functions with the largest improvement on the discus function where the expected runtime is more than halved. On the other hand, no statistically significant adverse effects can be observed

On the Impact of a Small Initial Population Size in the IPOP Active CMA-ES with Mirrored Mutations on the Noiseless BBOB Testbed

International audienceActive Covariance Matrix Adaptation and Mirrored Mutations have been independently proposed as improved variants of the well-known optimization algorithm Covariance Matrix Adaptation Evolution Strategy (CMA-ES) for numerical optimization. This paper investigates the impact of the algorithm's population size when both active covariance matrix adaptation and mirrored mutation are used in the CMA-ES. To this end, we compare the CMA-ES with standard population size $\lambda$, i.e., $\lambda = 4 + \lfloor 3\log(D) \rfloor$ with a version with half this population size where $D$ is the problem dimension