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Variable order structures model situations in which the comparison between two points depends on a point-to-cone map. In this paper, inexact projected gradient methods for solving smooth constrained vector optimization problems on variable ordered spaces are presented. It is shown that every accumulation point of the generated sequences satisfies the first-order necessary optimality condition. Moreover, under suitable convexity assumptions for the objective function, it is proved that all accumulation points of any generated sequences are weakly efficient points. The convergence results are also derived in the particular case in which the problem is unconstrained and even if inexact 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 coincides with the image of the objective function. 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. 

In this paper, we generalize the classical extragradient algorithm for solving variational inequality problems by utilizing nonzero normal vectors of the feasible set. In particular, conceptual algorithms are proposed with two different linesearchs. We then establish convergence results for these algorithms under mild assumptions. Our study suggests that nonzero normal vectors may significantly improve convergence if chosen appropriately.