In this paper, we study the asymptotic behavior for multi-scale stochastic
differential equations driven by L\'evy processes. The optimal strong
convergence order 1/2 is obtained by studying the regularity estimates for the
solution of Poisson equation with polynomial growth coefficients, and the
optimal weak convergence order 1 is got by using the technique of Kolmogorov
equation. The main contribution is that the obtained results can be applied to
a class of multi-scale stochastic differential equations with monotonicity
coefficients, as well as the driven processes can be the general L\'evy
processes, which seems new in the existing literature.Comment: 39 pages. To appear in Potential Analysi