  Quadratic unconstrained binary optimization (QUBO) solvers can be applied to
design an optimal structure to avoid resonance. QUBO algorithms that work on a
classical or quantum device have succeeded in some industrial applications.
However, their applications are still limited due to the difficulty of
transforming from the original optimization problem to QUBO. Recently,
black-box optimization (BBO) methods have been proposed to tackle this issue
using a machine learning technique and a Bayesian treatment for combinatorial
optimization. We employed the BBO methods to design a printed circuit board for
resonance avoidance. This design problem is formulated to maximize natural
frequency and simultaneously minimize the number of mounting points. The
natural frequency, which is the bottleneck for the QUBO formulation, is
approximated to a quadratic model in the BBO method. We demonstrated that BBO
using a factorization machine shows good performance in both the calculation
time and the success probability of finding the optimal solution. Our results
can open up QUBO solvers' potential for other applications in structural
designs.
