  We propose a new method for solving binary optimization problems under
inequality constraints using a quantum annealer. To deal with inequality
constraints, we often use slack variables, as in previous approaches. When we
use slack variables, we usually conduct a binary expansion, which requires
numerous physical qubits. Therefore, the problem of the current quantum
annealer is limited to a small scale. In this study, we employ the alternating
direction method of multipliers. This approach allows us to deal with various
types using constraints in the current quantum annealer without slack
variables. To test the performance of our algorithm, we use quadratic knapsack
problems (QKPs). We compared the accuracy obtained by our method with a
simulated annealer and the optimization and sampling mode of a D-Wave machine.
As a result of our experiments, we found that the sampling mode shows the best
accuracy. We also found that the computational time of our method is faster
than that of the exact solver when we tackle various QKPs defined on dense
graphs.
