Discrete Dynamical System Approaches for Boolean Polynomial Optimization

In this article, we discuss the numerical solution of Boolean polynomial programs by algorithms borrowing from methods for differential equations, namely Houbolt scheme, Lie and a Runge-Kutta scheme. We first introduce quartic penalty functional (of Ginzburg-Landau type) to approximate program continuous one prove some convergence results as parameter $$\varepsilon $$ converges 0. also that, under reasonable assumptions, distance between local minimizers penalized problem set $$\{\pm 1\}^n$$ is order $$O(\sqrt{n}\varepsilon )$$ . Next, problem, these relying on Houbolt, schemes, classical ordinary or partial equations. performed experiments investigate impact various parameters algorithms. have tested our ODE approaches compared with nonlinear optimization solver IPOPT quadratic binary formulation approach (QB-G) well an exhaustive method using parallel computing techniques. The datasets (including small large-scale randomly generated synthetic general problems, heterogeneous MQLib benchmark dataset Max-Cut Quadratic Unconstrained Binary Optimization (QUBO) problems) show good performances approaches. As result, often converge faster than other better integer solutions program.