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Discrete Dynamical System Approaches for Boolean Polynomial Optimization

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.