  In this article, we discuss the numerical solution of Boolean polynomial
programs by algorithms borrowing from numerical methods for differential
equations, namely the Houbolt scheme, the Lie scheme, and a Runge-Kutta scheme.
We first introduce a quartic penalty functional (of Ginzburg-Landau type) to
approximate the Boolean program by a continuous one and prove some convergence
results as the penalty parameter $\varepsilon$ converges to $0$. We prove also
that, under reasonable assumptions, the distance between local minimizers of
the penalized problem and the set $\{\pm1\}^n$ is of order
$O(\sqrt{n}\varepsilon)$. Next, we introduce algorithms for the numerical
solution of the penalized problem, these algorithms relying on the Houbolt, Lie
and Runge-Kutta schemes, classical methods for the numerical solution of
ordinary or partial differential equations. We performed numerical experiments
to investigate the impact of various parameters on the convergence of the
algorithms. We have tested our ODE approaches and compared with the classical
nonlinear optimization solver IPOPT and a quadratic binary formulation approach
(QB-G) as well as an exhaustive method using parallel computing techniques. The
numerical results on various datasets (including small and large-scale randomly
generated synthetic datasets of general Boolean polynomial optimization
problems, and a large-scale heterogeneous MQLib benchmark dataset of Max-Cut
and Quadratic Unconstrained Binary Optimization (QUBO) problems) show good
performances for our ODE approaches. As a result, our ODE algorithms often
converge faster than the other compared methods to better integer solutions of
the Boolean program.
