The interior-point method (IPM) has become the workhorse method for nonlinear programming. The performance of IPM is directly related to the linear solver employed to factorize the Karush--Kuhn--Tucker (KKT) system at each iteration of the algorithm. When solving large-scale nonlinear problems, state-of-the art IPM solvers rely on efficient sparse linear solvers to solve the KKT system. Instead, we propose a novel reduced-space IPM algorithm that condenses the KKT system into a dense matrix whose size is proportional to the number of degrees of freedom in the problem. Depending on where the reduction occurs we derive two variants of the reduced-space method: linearize-then-reduce and reduce-then-linearize. We adapt their workflow so that the vast majority of computations are accelerated on GPUs. We provide extensive numerical results on the optimal power flow problem, comparing our GPU-accelerated reduced space IPM with Knitro and a hybrid full space IPM algorithm. By evaluating the derivatives on the GPU and solving the KKT system on the CPU, the hybrid solution is already significantly faster than the CPU-only solutions. The two reduced-space algorithms go one step further by solving the KKT system entirely on the GPU. As expected, the performance of the two reduction algorithms depends intrinsically on the number of available degrees of freedom: their performance is poor when the problem has many degrees of freedom, but the two algorithms are up to 3 times faster than Knitro as soon as the relative number of degrees of freedom becomes smaller.