Andforanyα>0,theLaplaciansatisfies Gλ c1 2α Gλ+2Md whereM =c23α/c1+c22. 2. IfG (x)=kg (x)k2 isC1,then k G (x)k c2kg (x)k,g (x) > G (x) c1G (x). Proof. Claim1.Note Gλ =2 2Fλgλandthat 1 2 c1I 2Fλ= mX

As a remark, using the Krylov-Bogoliubov existence theorem (see Corollary 11.8 of [6]), fixed points to(4)exist as long as one can show{ρt,t 0}istight. The learning rate is set differently foreachtask. Obviously, the HV indicator (Eq.(10)) can also be used as an objective function for optimizing solution sets. For example, [25, 7] greedily add new points to obtain the highest expected HV improvement. However, the landscape of the HV indicator is piece-wise constant (similar to the 0-1 loss in classification) and is difficult to optimize with gradient descent. Particularly, for all the dominated points inthe solution set, their gradient iszero.