  Solving combinatorial optimization problems on current noisy quantum devices
is currently being advocated for (and restricted to) binary polynomial
optimization with equality constraints via quantum heuristic approaches. This
is achieved using, e.g., the variational quantum eigensolver (VQE) and the
quantum approximate optimization algorithm (QAOA). We present a
decomposition-based approach to extend the applicability of current approaches
to "quadratic plus convex" mixed binary optimization (MBO) problems, so as to
solve a broad class of real-world optimization problems. In the MBO framework,
we show that the alternating direction method of multipliers (ADMM) can split
the MBO into a binary unconstrained problem (that can be solved with quantum
algorithms), and continuous constrained convex subproblems (that can be solved
cheaply with classical optimization solvers). The validity of the approach is
then showcased by numerical results obtained on several optimization problems
via simulations with VQE and QAOA on the quantum circuits implemented in
Qiskit, an open-source quantum computing software development framework.
