Multiobjective abstract schemas for descent algorithms: solving conflicts of objectives via a variational rationality perspective

Celebrated[Luo ZQ, Tseng P. Error bounds and convergence analysis of feasible descent methods: a general approach. Ann Oper Res. 1993;46(1):157–178], abstract schema appeared recently as an important tool in mathematics, including two conditions that drive a lot of descent algorithms ([Attouch H, Bolte J, Svaiter B. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math Program. 2013;137(Ser A):91–129]). The first (second) condition reflects a sufficient descent condition (a relative inexact optimality condition). Despite their great merits, they cannot be applied easily in multiobjective behavioural sciences because, in the context of the recent (VR) variational rationality approach of stay and change human dynamics (Soubeyran [Soubeyran A. Variational rationality, a theory of individual stability and change: worthwhile and ambidextry behaviours. GREQAM, Aix Marseille University. Preprint, 2009.]–[Soubeyran A. Variational rationality: Self regulation success as a succession of worthwhile moves that make sufficient progress. AMSE, Aix-Marseille University. Preprint 2021e.]), while the first condition is clear (but requires, first, to be generalized to asymmetric distances), the second condition is not. In this paper, we give a new formulation of the second condition that is much more intuitive in multiobjective behavioural sciences, making, as the VR approach urged to do, a clear difference between crude motivation and the force of motivation. This new condition says that the more you do the same thing (habit formation), the less the force of your motivation to change. This is an important reinforcement condition, traditional in habit forming and breaking processes.