Componentwise versus global approaches
to nonsmooth multiobjective optimization
Giancarlo Bigi
Summary
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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