Variational rationality, variational principles and the existence of traps in a changing environment

Abstract

This paper has two aspects. Mathematically, in the context of global optimization, it provides the existence of an optimum of a perturbed optimization problem that generalizes the celebrated Ekeland variational principle and equivalent formulations (Caristi, Takahashi), whenever the perturbations need not satisfy the triangle inequality. Behaviorally, it is a con-tinuation of the recent variational rationality approach of stay (stop) and change (go) human dynamics. It gives sufficient conditions for the existence of traps in a changing environment. In this way it emphasizes even more the striking correspondence between variational analysis in mathematics and variational rationality in psychology and behavioral sciences.

Robust Ekeland variational principles. Application to the formation and stability of partnerships

“…Inspired by [9,14], we are going to obtain general forms of the robust Ekeland variational principle in terms of set-valued EVP with variable preferences, uncertain parameters and changing weights given to vectorial perturbation functions. Motivated by the notion of sequentially lower monotonicity for vector-valued and set-valued maps; see [10,14,15,50], we extend this notion for set-valued bimaps with variable ordering.…”

“…Ekeland's variational principle [1] (briefly, denoted by EVP) is a well-known theorem that has several versions, extensions and applications in nonlinear analysis, optimization, psychology and the behavioural sciences; see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein. Since EVP provides the existence of a strict minimum of a perturbed lower semicontinuous function on complete metric spaces types criteria of set-valued optimization are not comparable [26,29], there is no direct comparison between the two types of EVP extensions to set-valued maps.…”

“…Section 3 is devoted to the mathematical part. Following Fakhar et al [9] and Qiu et al [14], and driven by behavioural applications, we give new formulations of the robust Ekeland variational principle in terms of set-valued EVP with variable preferences, uncertain parameters and changing weights given to vectorial perturbation functions. Section 4 is devoted to modelling the formation and stability of a partnership in a changing, uncertain and complex environment in the context of the variational rationality approach of stop, continue and go human dynamics.…”

Set-valued variational principles. When migration improves quality of life

“…The latter applications are based on the variational rationality approach in behavioral sciences, initiated by Soubeyran in [29,30]. More recently, some researchers obtained several extensions of the EVP with applications in behavioral sciences; see, e.g., [7,9,10,11,15,16,24,27] and the references therein. However, there are "two types of EVP in the case of set-valued mappings": the vector criterion (see, e.g., [2,7,8,9,13,14,19,23]) and the set criterion (see, e.g., [3,4,20,21,22,28]).…”

“…The second criterion is to find a strict minimum of a perturbed set-valued mapping with respect to the set order relation in the class of all nonempty subsets of the image space. Following Fakhar et al [15,16] and Qiu et al [24], and driven by behavioral science applications, we give new formulations of set-valued versions of the EVP by using lower and upper set less relations with variable preferences. An application is given in relation to migration problems and the quality of life in the framework of the variational rationality approach of stay (continue) and change (stop and start) human dynamics [31,32,33,34].…”