Trade-off invariance for weighted scalarizations in multi-objective optimization

Abstract

We consider weighted-sum scalarizations for an abstract multi-objective minimization problem defined by the vector-valued map $U\ni u\mapsto \big( f_1(u),\ldots, f_N(u)\big)$, where $U$ is an arbitrary nonempty set and no topology, convexity, compactness, or lower semicontinuity assumption is imposed. Using the open simplex as parameter space for positive weights, we show that the Trade-off Invariance Principle for scalarizations yields a generic uniqueness property in the objective space. Namely, for almost every weight vector, all minimizers of the corresponding weighted-sum scalarization have the same objective vector, although the minimizers themselves need not be unique. Moreover, excluding again a null-measure subset, all minimizing sequences determine the same limiting objective vector, independently of the chosen sequence. We also give a geometric interpretation of these results in the attainable objective set: for almost every positive weight vector, the scalarization exposes at most one nondominated point. Furthermore, minimizing sequences determine at most one asymptotically exposed objective vector in the closure of the attainable set.