  In this work, we study a classic robust design problem in two-hop one-way
relay system. We are particularly interested in the scenario where channel
uncertainty exists in both the transmitter-to-relay and relay-to-receiver
links. By considering the problem design that minimizes the average
amplify-and-forward power budget at the relay side while satisfying SNR outage
requirements, an outage-constrained robust design problem involving quartic
perturbations is formulated to guarantee the robustness during transmission.
This problem is in general difficult as it involves constraints on the tail
probability of a high-order polynomial. Herein, we resort to moment inequality
and Bernstein-type inequality to tackle this problem, which provide convex
restrictions, or safe approximations, of the original design. We also analyze
the relative tightness of the two safe approximations for a quadratic
perturbation-based outage constrained problem. Our analysis shows that the
Bernstein-type inequality approach is less conservative than the moment
inequality approach when the outage rate is within some prescribed regime. To
our best knowledge, this is the first provable tightness result for these two
safe approximations. Our numerical simulations verify the superiority of the
robust design and corroborate the tightness results.
