In this paper, we show how to use the GPU to parallelize a precise instruction scheduling algorithm that is based on Ant Colony Optimization (ACO). ACO is a nature-inspired intelligent-search technique that has been used to compute precise solutions to NP-hard problems in operations research (OR). Such intelligent-search techniques were not used in the past to solve NP-hard compiler optimization problems, because they require substantially more computation than the heuristic techniques used in production compilers. In this work, we show that parallelizing such a compute-intensive technique on the GPU makes using it in compilation reasonably practical. The register-pressure-aware instruction scheduling problem addressed in this work is a multi-objective optimization problem that is significantly more complex than the problems that were previously solved using parallel ACO on the GPU. We describe a number of techniques that we have developed to efficiently parallelize an ACO algorithm for solving this multi-objective optimization problem on the GPU. The target processor is also a GPU. Our experimental evaluation shows that parallel ACO-based scheduling on the GPU runs up to 27 times faster than sequential ACO-based scheduling on the CPU, and this leads to reducing the total compile time of the rocPRIM benchmarks by 21%. ACO-based scheduling improves the execution-speed of the compiled benchmarks by up to 74% relative to AMD's production scheduler. To the best of our knowledge, our work is the first successful attempt to parallelize a compiler optimization algorithm on the GPU.