The discovery of new materials in unexplored chemical spaces necessitates quick and accurate prediction of thermodynamic stability, often assessed using density functional theory (DFT), and efficient search strategies. Here, we develop a new approach to finding stable inorganic functional materials. We start by defining an upper bound to the fully-relaxed energy obtained via DFT as the energy resulting from a constrained optimization over only cell volume. Because the fractional atomic coordinates for these calculations are known a priori, this upper bound energy can be quickly and accurately predicted with a scale-invariant graph neural network (GNN). We generate new structures via ionic substitution of known prototypes, and train our GNN on a new database of 128,000 DFT calculations comprising both fully-relaxed and volume-only relaxed structures. By minimizing the predicted upper-bound energy, we discover new stable structures with over 99% accuracy (versus DFT). We demonstrate the method by finding promising new candidates for solid-state battery (SSB) electrolytes that not only possess the required stability, but also additional functional properties such as large electrochemical stability windows and high conduction ion fraction. We expect this proposed framework to be directly applicable to a wide range of design challenges in materials science.