  Given a finite set $N$ of feasible points of a multi-objective optimization
(MOO) problem, the search region corresponds to the part of the objective space
containing all the points that are not dominated by any point of $N$, i.e. the
part of the objective space which may contain further nondominated points. In
this paper, we consider a representation of the search region by a set of tight
local upper bounds (in the minimization case) that can be derived from the
points of $N$. Local upper bounds play an important role in methods for
generating or approximating the nondominated set of an MOO problem, yet few
works in the field of MOO address their efficient incremental determination. We
relate this issue to the state of the art in computational geometry and provide
several equivalent definitions of local upper bounds that are meaningful in
MOO. We discuss the complexity of this representation in arbitrary dimension,
which yields an improved upper bound on the number of solver calls in
epsilon-constraint-like methods to generate the nondominated set of a discrete
MOO problem. We analyze and enhance a first incremental approach which operates
by eliminating redundancies among local upper bounds. We also study some
properties of local upper bounds, especially concerning the issue of redundant
local upper bounds, that give rise to a new incremental approach which avoids
such redundancies. Finally, the complexities of the incremental approaches are
compared from the theoretical and empirical points of view.
