Convergences for robust bilevel polynomial programmes with applications

In this paper, we consider a bilevel polynomial optimization problem, where the constraint functions of both the upper-level and lower-level problems involve uncertain parameters. We employ the deterministic robust optimization approach to examine the bilevel polynomial optimization problem under data uncertainties by providing lower bound approximations and convergences of sum-of-squares (SOS) relaxations for the robust bilevel polynomial optimization problem. More precisely, we show that under the convexity of the lower-level problem and either the boundedness of the feasible set or the coercivity of the objective function, the global optimal values of SOS relaxation problems are lower bounds of the global optimal value of the robust bilevel polynomial problem and they converge to this global optimal value when the degrees of SOS polynomials in the relaxation problems tend to infinity. Moreover, an application to an electric vehicle charging scheduling problem with renewable energy sources demonstrates that using the proposed SOS relaxation schemes, we obtain more stable optimal values than applying a direct solution approach as the SOS relaxations are capable of solving these models involving data uncertainties in dynamic charging price and weather conditions.