Approximate weak efficiency of the set-valued optimization problem with variable ordering structures

In locally convex spaces, we introduce the new notion of approximate weakly efficient solution of the set-valued optimization problem with variable ordering structures (in short, SVOPVOS) and compare it with other kinds of solutions. Under the assumption of near \(\mathcal {D}(\cdot )\)-subconvexlikeness, we establish linear scalarization theorems of (SVOPVOS) in the sense of approximate weak efficiency. Finally, without any convexity, we obtain a nonlinear scalarization theorem of (SVOPVOS). We also present some examples to illustrate our results.