Nanoscale variations in composition arising from the competition between chemical mixing effects and elastic relaxation can substantially influence the electronic and optical properties of self-assembled alloy quantum dots. Using a combination of finite element and quadratic programming optimization methods, we have developed an efficient technique to compute the equilibrium composition profiles in strained quantum dots. We find that the composition profiles depend strongly on the morphological features such as the slopes and curvatures of their surfaces and the presence of corners and edges as well as the ratio of the strain and chemical mixing energy densities. More generally, our approach provides a means to quantitatively model the interplay among the composition variations, the temperature, the strain, and the shapes of small-scale lattice-mismatched structures.