Unistable polyhedra are in equilibrium on only one of their faces. The smallest known homogeneous unistable polyhedron to date has 18 faces. Using a new optimization algorithm, we have found a unistable polyhedron with only 14 faces, which we believe to be a lower bound. Despite the simplicity of the formulation, computers were never successfully used for solving this problem due to the seemingly insurmountable dimensionality of the underlying mathematical apparatus. We introduce new optimization approaches designed to overcome the problem's intractability and discuss its significance to other application areas. We also mathematically prove the unistable property of the found bodies using rational arithmetic. Surprisingly enough, all of our computer-generated unistable polyhedra look similar to the human eye, providing important insights into the nature of the problem.