In memory of Asen L. Dontchev. Abstract. In this paper, we introduce a topology on the power set P(Z) of a partially ordered normed space Z from which we derive a topological convergence on P(Z) along with new concepts of continuity and semicontinuity for set-valued mappings. Our goal is to propose an appropriate framework to address set optimization problems involving set relations based on a cone ordering. Taking advantage of this new setting, we establish several results regarding the well-posedness of set-valued optimization problems that are consistent with the state-of-the-art.

The aim of this study was to model the behavior of particles with aerodynamic diameter lower or equal to 10 µm (PM10) in the Caribbean area according to African dust seasonality. To carry out this study, PM10 measurement from Guadeloupe (GPE) and Puerto Rico (PR) between 2006 and 2010 were used. Firstly, the missing data issues were addressed using algorithms that we elaborated. Thereafter, the coupled SARIMA-GARCH (Seasonal Autoregressive Integrated Moving Average and Generalized Autoregressive Conditional Heteroscedastic) model was developed and compared to PM10 empirical data. The SARIMA process is representative of the main PM10 sources, while the heteroskedasticity is also taken into account by the GARCH process. In this framework, PM10 data from GPE and PR are decomposed into the sum of the background atmosphere (Bt = anthropogenic activities + marine aerosol), African dust seasonality (St = mineral dust), and extreme events processes (Ct). Akaike’s information criterion (AIC) helped us to choose the best model. Forecast evaluation indexes such as the Mean Absolute Percentage Error (MAPE), the Mean Absolute Scale Error (MASE), and Theil’s U statistic provided significant results. Specifically, the MASE and U values were found to be almost zero. Thus, these indexes validated the forecasts of the coupled SARIMA-GARCH model. To sum up, the SARIMA-GARCH combination is an efficient tool to forecast PM10 behavior in the Caribbean area.