  Variable order structures model situations in which the comparison between
two points depends on a point-to-cone map. In this paper, an inexact projected
gradient method for solving smooth constrained vector optimization problems on
variable ordered spaces is presented. It is shown that every accumulation point
of the generated sequence satisfies the first order necessary optimality
condition. The convergence of all accumulation points to a weakly efficient
point is established under suitable convexity assumptions for the objective
function. The convergence results are also derived in the particular case in
which the problem is unconstrained and if exact directions are taken as descent
directions. Furthermore, we investigate the application of the proposed method
to optimization models where the domain of the variable order map and the
objective function are the same. In this case, similar concepts and convergence
results are presented. Finally, some computational experiments designed to
illustrate the behavior of the proposed inexact methods versus the exact ones
(in terms of CPU time) are performed.
