Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming

AbstractSample average approximation (SAA) is one of the most popular methods for solving stochastic optimization and equilibrium problems. Research on SAA has been mostly focused on the case when sampling is independent and identically distributed (iid) with exceptions (Dai et al. (2000) [9], Homem-de-Mello (2008) [16]). In this paper we study SAA with general sampling (including iid sampling and non-iid sampling) for solving nonsmooth stochastic optimization problems, stochastic Nash equilibrium problems and stochastic generalized equations. To this end, we first derive the uniform exponential convergence of the sample average of a class of lower semicontinuous random functions and then apply it to a nonsmooth stochastic minimization problem. Exponential convergence of estimators of both optimal solutions and M-stationary points (characterized by Mordukhovich limiting subgradients (Mordukhovich (2006) [23], Rockafellar and Wets (1998) [32])) are established under mild conditions. We also use the unform convergence result to establish the exponential rate of convergence of statistical estimators of a stochastic Nash equilibrium problem and estimators of the solutions to a stochastic generalized equation problem

A characterization of nonemptiness and boundedness of the solution set for set-valued vector equilibrium problems via scalarization and stability results

International audienceAttitude is a key concept in social psychology. The paper presents a novel agent-based model to simulate attitude formation by combining a rational and an emotional components based on cognitive, psychological and social theories. Individuals of the artificial population perceive actions taken by actors such as government or brands, they form an attitude toward them and also communicate the events through a social network. The model outputs are first studied through a functional analysis in which some unique macroscopic behaviors have emerged such as the impact of social groups, the resistance of the population toward disinformation campaigns or the social pressure. We then applied our model on a real world scenario depicting the effort of French Forces in their stabilization operations in Kapisa (Afghanistan) between 2010 and 2012. We calibrated the model parameters based on this scenario and the results of opinion polls that were conducted in the area during the same period about the sentiment of the population toward the Forces. Our model was able to reproduce polls results with a global error under 3%. Based on these results, we show the different dynamics tendencies that emerged among the population by applying a non-supervised classification algorithm