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In this paper, we first derive Milstein schemes for an interacting particlesystem associated with point delay McKean-Vlasov stochastic differentialequations (McKean-Vlasov SDEs), possibly with a drift term exhibitingsuper-linear growth in the state component. We prove strong convergence oforder one and moment stability, making use of techniques from variationalcalculus on the space of probability measures with finite second order moments.Then, we introduce an antithetic multilevel Milstein scheme, which leads tooptimal complexity estimators for expected functionals of solutions to delayMcKean-Vlasov equations without the need to simulate L\'evy areas.

Milstein schemes and antithetic multilevel Monte Carlo sampling for delay McKean-Vlasov equations and interacting particle systems