Supplementary File of “ Dynamic Multi-Objectives Optimization with a Changing Number of Objectives ”

Proof. Let us consider the scenario of increasing the number of objectives at first, where we prove the theorem by contradiction. At time step t1, we assume that x 1 and x are in PSt1 . Accordingly, F(x , t1) and F(x , t1) are in PFt1 . At time step t2, we increase the number of objectives by one, i.e., m(t2) = m(t1)+ 1. Assume that x 1 is still in PSt2 whereas x 2 is not, thus we have x t2 x . In other words, ∀i ∈ {1, · · · ,m(t1),m(t2)} (i.e., ∀i ∈ {1, · · · ,m(t1),m(t1) + 1}), fi(x , t2) ≤ fi(x , t2); and ∃j ∈ {1, · · · ,m(t1),m(t1)+1}, fj(x , t2) < fj(x , t2). This contradicts the assumption that x 1 and x are non-dominated from each other at time step t1. Then, we conclude that PFt1 is a subset of PFt2 when increasing the number of objectives. Now let us consider the scenario of decreasing the number of objectives. At time step t1, we assume that x 1 and x are in PSt1 . Accordingly, F(x , t1) and F(x , t1) are in PFt1 . Furthermore, we assume that ∀i ∈ {1, · · · ,m(t1)− 1}, we have fi(x , t1) ≤ fi(x , t1) and fm(t1)(x , t1) > fm(t1)(x , t1). At time step t2, we decrease the number of objectives by one, i.e., m(t2) = m(t1) − 1. If fm(t1) is removed at time step t2, we can derive that x 1 t2 x . That is to say x is not in PFt2 . On the other hand, if fi, where i ∈ {1, · · · ,m(t1) − 1}, is removed at time step t2, we can derive that x 1 and x are still non-dominated from each other. In other words, x and x are still in PFt2 . All in all, we conclude that PFt1 is a superset of PFt2 .