  We present a hybrid classical-quantum framework based on the Frank-Wolfe
algorithm, Q-FW, for solving quadratic, linearly-constrained, binary
optimization problems on quantum annealers (QA). The computational premise of
quantum computers has cultivated the re-design of various existing vision
problems into quantum-friendly forms. Experimental QA realizations can solve a
particular non-convex problem known as the quadratic unconstrained binary
optimization (QUBO). Yet a naive-QUBO cannot take into account the restrictions
on the parameters. To introduce additional structure in the parameter space,
researchers have crafted ad-hoc solutions incorporating (linear) constraints in
the form of regularizers. However, this comes at the expense of a
hyper-parameter, balancing the impact of regularization. To date, a true
constrained solver of quadratic binary optimization (QBO) problems has lacked.
Q-FW first reformulates constrained-QBO as a copositive program (CP), then
employs Frank-Wolfe iterations to solve CP while satisfying linear (in)equality
constraints. This procedure unrolls the original constrained-QBO into a set of
unconstrained QUBOs all of which are solved, in a sequel, on a QA. We use
D-Wave Advantage QA to conduct synthetic and real experiments on two important
computer vision problems, graph matching and permutation synchronization, which
demonstrate that our approach is effective in alleviating the need for an
explicit regularization coefficient.
