A brief but comprehensive review of the averaged Hausdorff distances that have recently been introduced as quality indicators in multi-objective optimization problems (MOPs) is presented. First, we introduce all the necessary preliminaries, definitions, and known properties of these distances in order to provide a stat-of-the-art overview of their behavior from a theoretical point of view. The presentation treats separately the definitions of the $(p,q)$ -distances ${GD}_{p,q}$ , ${IGD}_{p,q}$ , and ${\Delta }_{p,q}$ for finite sets and their generalization for arbitrary measurable sets that covers as an important example the case of continuous sets. Among the presented results, we highlight the rigorous consideration of metric properties of these definitions, including a proof of the triangle inequality for distances between disjoint subsets when $p,q⩾1$ , and the study of the behavior of associated indicators with respect to the notion of compliance to Pareto optimality. Illustration of these results in particular situations are also provided. Finally, we discuss a collection of examples and numerical results obtained for the discrete and continuous incarnations of these distances that allow for an evaluation of their usefulness in concrete situations and for some interesting conclusions at the end, justifying their use and further study.

The analysis of synoptic satellite data of total chlorophyll-a (Chl-a) and the environmental drivers that influence nutrient and light availability for phytoplankton growth allows us to understand the spatio-temporal variability of phytoplankton biomass. In the Panama Bight Tropical region (PB; 1–9°N, 79–84°W), the spatial distribution of Chl-a is mostly related to the seasonal wind patterns and the intensity of localized upwelling centers. However, the association between the Chl-a and different physical variables and nutrient availability is still not fully assessed. In this study, we evaluate the relationship between the Chl-a and multiple physical (wind, Ekman pumping, geostrophic circulation, mixed layer depth, sea level anomalies, river discharges, sea surface temperature, and photosynthetically available radiation) and chemical (nutrients) drivers in order to explain the spatio-temporal Chl-a variability in the PB. We used satellite data of Chl-a and physical variables, and a re-analysis of a biogeochemical product for nutrients (2002–2016). Our results show that at the regional scale, the Chl-a varies seasonally in response to the wind forcing and sea surface temperature. However, in the coastal areas (mainly Gulf of Panama and off central-southern Colombia), the maximum non-seasonal Chl-a values are found in association with the availability of nutrients by river discharges, localized upwelling centers and the geostrophic circulation field. From this study, we infer that the interplay among these physical-chemical drivers is crucial for supporting the phytoplankton growth and the high biodiversity of the PB region.

The Hausdorff distance is a widely used tool to measure the distance between different sets. For the approximation of certain objects via stochastic search algorithms this distance is, however, of limited use as it punishes single outliers. As a remedy in the context of evolutionary multi-objective optimization (EMO), the averaged Hausdorff distance ${\Delta }_p$ has been proposed that is better suited as an indicator for the performance assessment of EMO algorithms since such methods tend to generate outliers. Later on, the two-parameter indicator ${\Delta }_{p,q}$ has been proposed for finite sets as an extension to ${\Delta }_p$ which also averages distances, but which yields some desired metric properties. In this paper, we extend ${\Delta }_{p,q}$ to a continuous function between general bounded subsets of finite measure inside a metric measure space. In particular, this extension applies to bounded subsets of ${\mathbb{R}}^k$ endowed with the Euclidean metric, which is the natural context for EMO applications. We show that our extension preserves the nice metric properties of the finite case, and finally provide some useful numerical examples that arise in EMO.

A Generalization of the Averaged Hausdorff Distance

In this paper we study compactness of subsets of grand Lebesgue spaces (also known as Iwaniec–Sbordone spaces) and grand Sobolev spaces. Namely, we prove a Kolmogorov–Riesz compactness theorem for grand Lebesgue spaces and the corresponding version of a result of Sudakov. In addition, the validity of a Rellich–Kondrachov type compact embedding is shown to hold in the context of grand Sobolev spaces.