Resistance to pharmacological treatments, such as to antibiotics or cancer therapeutics, ablates the efficacy of drugs and is a major public health challenge. Accurate, prospective prediction of resistance mutations would allow for design of drugs less susceptible to conferred resistance. Here we report RESISTOR--a novel structure- and sequence-based algorithm for predicting resistance mutations. RESISTOR optimizes over four objectives to find the Pareto frontier of these resistance-causing criteria. The first two axes of optimization are provable approximations to the relative change in binding affinity (ΔKa) of a drug and endogenous ligand upon a protein's mutation. These ΔKa predictions are made by the provable thermodynamic- and ensemble-based multistate computational protein design algorithm K* after an initial sequence filter using the multistate design algorithm COMETS from the computational protein design software OSPREY. By virtue of pruning using COMETS, RESISTOR inherits the empirical sublinearlity characteristics of the COMETS sequence search, rendering RESISTOR, to our knowledge, the first provable structure-based resistance prediction algorithm that is sublinear in the size of the sequence space. This is important because the size of sequence space is exponential in the number of residue positions that can mutate to confer resistance. On the third axis RESISTOR uses empirically derived mutational signatures to determine the probability that a particular single- or double-point mutation will occur in a given context. The fourth axis of optimization is over "mutational hotspots", or those locations in the protein where multiple amino acids are predicted to confer resistance. As validation of the algorithm, we applied RESISTOR to a number of tyrosine kinase inhibitors used to treat lung adenocarcinoma and melanoma through inhibition of EGFR or B-Raf kinase activity. In so doing, we searched over a set of 1257 sequences in EGFR and 1214 sequences in B-Raf with an average conformation space size of ~5.9×1010, using experimental structures when available and docked complexes when not. This search generated a set of predicted resistance mutations that we compared to known resistance mutations in EGFR and report herein that RESISTOR correctly identified eight clinically significant resistance mutations, including the "gatekeeper" T790M mutation to erlotinib and gefitinib and five known resistance mutations to osimertinib. This demonstrates that by exploiting the wealth of structural and sequence data available in the form of molecular structures and mutational signatures, RESISTOR is a general method for predicting resistance mutations that can be applied to a wide variety of cancer, antimicrobial, antiviral and antifungal drug targets. RESISTOR is available as part of OSPREY on GitHub and is free and open source software.