Componentwise versus global approaches
to nonsmooth multiobjective optimization

Giancarlo Bigi


Nonsmooth multiobjective optimization has received much attention in the past years; in particular, many first order optimality conditions have been achieved, relying on different approaches based on nonsmooth analysis tools.
Since these tools are quite simple and powerful for real-valued functions, a standard approach is to consider suitable weighted sums of the components of vector-valued functions and to exploit them to study optimality . However, such an approach inherits some well-known drawbacks of scalarization techniques. Another way to rely on nonsmooth tools for real-valued functions is just some kind of componentwise approach, that is considering generalized derivatives or some suitable subdifferentials of the components of the considered functions.
Relying on the concept of Kuratowski limit, a very different approach has been introduced by Thibault and then applied to study multiobjective optimization in some recent papers. It can be considered somehow a global one, since set-valued directional derivatives of vector-valued functions are introduced without relying on components.
The aim of this paper is twofold. First, we deepen the analysis of the latter approach, providing new optimality conditions; we also show that well-known tools for vector functions such as generalized Jacobians can be incorporated into this global approach, just relying on particular set-valued derivatives. Then, we discuss whether or not the global approach provides sharper results than the componentwise one. For the sake of simplicity, we focus our attention only on Dini-Hadamard type derivatives and Pareto minimization problems, where the constraint is given just as a set.

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