  Black-box optimization has potential in numerous applications such as
hyperparameter optimization in machine learning and optimization in design of
experiments. Ising machines are useful for binary optimization problems because
variables can be represented by a single binary variable of Ising machines.
However, conventional approaches using an Ising machine cannot handle black-box
optimization problems with non-binary values. To overcome this limitation, we
propose an approach for integer-variable black-box optimization problems by
using Ising/annealing machines and factorization machines in cooperation with
three different integer-encoding methods. The performance of our approach is
numerically evaluated with different encoding methods using a simple problem of
calculating the energy of the hydrogen molecule in the most stable state. The
proposed approach can calculate the energy using any of the integer-encoding
methods. However, one-hot encoding is useful for problems with a small size.
