Nontransitive indifference preferences in bi-multi-objective optimization problems

In this paper, we introduce and discuss a generalization of the classical multi-objective optimization problem employing pairs of functions. More specifically, this procedure is referred to as bi-multi-objective optimization. A justification of this general optimization procedure is presented, related both to multi-objective optimization under ambiguity concerning individual preferences and to Pareto optimality for a family of preferences with nontransitive indifference. Incidentally, the binary relation naturally associated to a bi-multi-objective optimization problem is represented by a finite bi-multi-utility, which generalizes the classical finite multi-utility representation to the nontransitive case. Finally, we present an important application in the context of Markowitz portfolio selection under ambiguity concerning both the vector of returns and the covariance matrix.