Many material and electronic properties of systems depend on the atomic configuration at the surface. This problem can be viewed as a mixed variable optimization problem for the chemical identity of the atoms as well as their spatial coordinates. One common technique used for determining surface structures is based on solving an inverse problem to match some known data. An example arises in the determination of nanostructures where a technique known as low energy electron diffraction (LEED) method is used. We will describe the use of pattern search methods and simplified physics surrogates for determining the surface structure of nanosystems. The pattern search methods have the property of being able to handle both continuous and categorical variables, which allows the simultaneous optimization of the atomic coordinates as well as the chemical identity.