On well-posedness in vector optimization

The purpose of this work is to give an overview on the world of the vector well-posed optimization problems, according to the scalar notion by Tykhonov, in a finite dimensional setting, mainly under convexity or generalized convexity assumptions and focusing on scalarization procedures. Two characterizations of well-posedness properties are presented: the first involving all global notions compared, under generalized convexity assumption, the second involving all pointwise concepts and employing a vector version of Ekeland's variational principle due to Araya . Focusing on the strongest pointwise notion presented, Dentcheva-Helbig well-posedness with respect to an efficient point, we consider the idea of “how many” convex problems will have solutions and also enjoy the property of being well-posed. On this topic, we present a density result, that is the possibility to approximate a well-posed problem with a sequence of well-posed problems considering the same constraints and the same features for the objective functions. An overview of the application of the previous theoretical results to game theory is presented. Studying well-posedness one is naturally led to consider perturbations of functions and sets, that is a notion of well-posedness in the extended sense. We propose a notion for which sufficient conditions under convexity requirements are established. We consider several approaches and following the classification proposed by Miglierina et al., we stress the geometrical features of the image set of each property thanks to some illustrative examples. Thus we are able to establish the hierchical structure characterizing pointwise and global concepts. Finally, we turn our attention on scalarization technique, linear and nonlinear, to say that a vector problem is well-posed if and only if an associate, or more then one, scalar problem satisfies a scalar stability requirement.